@ Copyright 2019-2024 Mark Owen
@ http://www.quinapalus.com
@ E-mail: qfp@quinapalus.com
@
@ This file is free software: you can redistribute it and/or modify
@ it under the terms of version 2 of the GNU General Public License
@ as published by the Free Software Foundation.
@
@ This file is distributed in the hope that it will be useful,
@ but WITHOUT ANY WARRANTY; without even the implied warranty of
@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
@ GNU General Public License for more details.
@
@ You should have received a copy of the GNU General Public License
@ along with this file. If not, see <http://www.gnu.org/licenses/> or
@ write to the Free Software Foundation, Inc., 51 Franklin Street,
@ Fifth Floor, Boston, MA 02110-1301, USA.
.syntax unified
.cpu cortex-m0plus
.thumb
@ exported symbols
.global qfp_fadd
.global qfp_fsub
.global qfp_fmul
.global qfp_fdiv
.global qfp_fcmp
.global qfp_fsqrt
.global qfp_float2int
.global qfp_float2fix
.global qfp_float2uint
.global qfp_float2ufix
.global qfp_int2float
.global qfp_fix2float
.global qfp_uint2float
.global qfp_ufix2float
.global qfp_int642float
.global qfp_fix642float
.global qfp_uint642float
.global qfp_ufix642float
.global qfp_fcos
.global qfp_fsin
.global qfp_ftan
.global qfp_fatan2
.global qfp_fexp
.global qfp_fln
.global qfp_dadd
.global qfp_dsub
.global qfp_dmul
.global qfp_ddiv
.global qfp_dsqrt
.global qfp_dcos
.global qfp_dsin
.global qfp_dtan
.global qfp_datan2
.global qfp_dexp
.global qfp_dln
.global qfp_dcmp
.global qfp_float2int64
.global qfp_float2fix64
.global qfp_float2uint64
.global qfp_float2ufix64
.global qfp_float2int_z
.global qfp_float2int64_z
.global qfp_double2int
.global qfp_double2fix
.global qfp_double2uint
.global qfp_double2ufix
.global qfp_double2int64
.global qfp_double2fix64
.global qfp_double2uint64
.global qfp_double2ufix64
.global qfp_double2int_z
.global qfp_double2int64_z
.global qfp_int2double
.global qfp_fix2double
.global qfp_uint2double
.global qfp_ufix2double
.global qfp_int642double
.global qfp_fix642double
.global qfp_uint642double
.global qfp_ufix642double
.global qfp_double2float
.global qfp_float2double
qfp_lib_start:
@ exchange r0<->r1, r2<->r3
xchxy:
push {r0,r2,r14}
mov r0,r1
mov r2,r3
pop {r1,r3,r15}
@ IEEE single in r0-> signed (two's complemennt) mantissa in r0 9Q23 (24 significant bits), signed exponent (bias removed) in r2
@ trashes r4; zero, denormal -> mantissa=+/-1, exponent=-380; Inf, NaN -> mantissa=+/-1, exponent=+640
unpackx:
lsrs r2,r0,#23 @ save exponent and sign
lsls r0,#9 @ extract mantissa
lsrs r0,#9
movs r4,#1
lsls r4,#23
orrs r0,r4 @ reinstate implied leading 1
cmp r2,#255 @ test sign bit
uxtb r2,r2 @ clear it
bls 1f @ branch on positive
rsbs r0,#0 @ negate mantissa
1:
subs r2,#1
cmp r2,#254 @ zero/denormal/Inf/NaN?
bhs 2f
subs r2,#126 @ remove exponent bias: can now be -126..+127
bx r14
2: @ here with special-case values
cmp r0,#0
mov r0,r4 @ set mantissa to +1
bpl 3f
rsbs r0,#0 @ zero/denormal/Inf/NaN: mantissa=+/-1
3:
subs r2,#126 @ zero/denormal: exponent -> -127; Inf, NaN: exponent -> 128
lsls r2,#2 @ zero/denormal: exponent -> -508; Inf, NaN: exponent -> 512
adds r2,#128 @ zero/denormal: exponent -> -380; Inf, NaN: exponent -> 640
bx r14
@ normalise and pack signed mantissa in r0 nominally 3Q29, signed exponent in r2-> IEEE single in r0
@ trashes r4, preserves r1,r3
@ r5: "sticky bits", must be zero iff all result bits below r0 are zero for correct rounding
packx:
lsrs r4,r0,#31 @ save sign bit
lsls r4,r4,#31 @ sign now in b31
bpl 2f @ skip if positive
cmp r5,#0
beq 11f
adds r0,#1 @ fiddle carry in to following rsb if sticky bits are non-zero
11:
rsbs r0,#0 @ can now treat r0 as unsigned
packx0:
bmi 3f @ catch r0=0x80000000 case
2:
subs r2,#1 @ normalisation loop
adds r0,r0
beq 1f @ zero? special case
bpl 2b @ normalise so leading "1" in bit 31
3:
adds r2,#129 @ (mis-)offset exponent
bne 12f @ special case: highest denormal can round to lowest normal
adds r0,#0x80 @ in special case, need to add 256 to r0 for rounding
bcs 4f @ tripped carry? then have leading 1 in C as required
12:
adds r0,#0x80 @ rounding
bcs 4f @ tripped carry? then have leading 1 in C as required (and result is even so can ignore sticky bits)
cmp r5,#0
beq 7f @ sticky bits zero?
8:
lsls r0,#1 @ remove leading 1
9:
subs r2,#1 @ compensate exponent on this path
4:
cmp r2,#254
bge 5f @ overflow?
adds r2,#1 @ correct exponent offset
ble 10f @ denormal/underflow?
lsrs r0,#9 @ align mantissa
lsls r2,#23 @ align exponent
orrs r0,r2 @ assemble exponent and mantissa
6:
orrs r0,r4 @ apply sign
1:
bx r14
5:
movs r0,#0xff @ create infinity
lsls r0,#23
b 6b
10:
movs r0,#0 @ create zero
bx r14
7: @ sticky bit rounding case
lsls r5,r0,#24 @ check bottom 8 bits of r0
bne 8b @ in rounding-tie case?
lsrs r0,#9 @ ensure even result
lsls r0,#10
b 9b
.align 2
.ltorg
@ signed multiply r0 1Q23 by r1 4Q23, result in r0 7Q25, sticky bits in r5
@ trashes r3,r4
mul0:
uxth r3,r0 @ Q23
asrs r4,r1,#16 @ Q7
muls r3,r4 @ L*H, Q30 signed
asrs r4,r0,#16 @ Q7
uxth r5,r1 @ Q23
muls r4,r5 @ H*L, Q30 signed
adds r3,r4 @ sum of middle partial products
uxth r4,r0
muls r4,r5 @ L*L, Q46 unsigned
lsls r5,r4,#16 @ initialise sticky bits from low half of low partial product
lsrs r4,#16 @ Q25
adds r3,r4 @ add high half of low partial product to sum of middle partial products
@ (cannot generate carry by limits on input arguments)
asrs r0,#16 @ Q7
asrs r1,#16 @ Q7
muls r0,r1 @ H*H, Q14 signed
lsls r0,#11 @ high partial product Q25
lsls r1,r3,#27 @ sticky
orrs r5,r1 @ collect further sticky bits
asrs r1,r3,#5 @ middle partial products Q25
adds r0,r1 @ final result
bx r14
.thumb_func
qfp_fcmp:
lsls r2,r0,#1
lsrs r2,#24
beq 1f
cmp r2,#0xff
bne 2f
1:
lsrs r0,#23 @ clear mantissa if NaN or denormal
lsls r0,#23
2:
lsls r2,r1,#1
lsrs r2,#24
beq 1f
cmp r2,#0xff
bne 2f
1:
lsrs r1,#23 @ clear mantissa if NaN or denormal
lsls r1,#23
2:
movs r2,#1 @ initialise result
eors r1,r0
bmi 4f @ opposite signs? then can proceed on basis of sign of x
eors r1,r0 @ restore y
bpl 1f
rsbs r2,#0 @ both negative? flip comparison
1:
cmp r0,r1
bgt 2f
blt 3f
5:
movs r2,#0
3:
rsbs r2,#0
2:
subs r0,r2,#0
bx r14
4:
orrs r1,r0
adds r1,r1
beq 5b
cmp r0,#0
bge 2b
b 3b
@ convert float to signed int, rounding towards 0, clamping
.thumb_func
qfp_float2int_z:
push {r14}
cmp r0,#0
blt 1f
bl qfp_float2int @ +ve or zero? then use rounding towards -Inf
cmp r0,#0
bge 2f
ldr r0,=#0x7fffffff
2:
pop {r15}
1:
lsls r0,#1 @ -ve: clear sign bit
lsrs r0,#1
bl qfp_float2uint @ convert to unsigned, rounding towards -Inf
cmp r0,#0
blt 1f
rsbs r0,#0
pop {r15}
1:
movs r0,#1
lsls r0,#31 @ 0x80000000
pop {r15}
.ltorg
@ convert float to signed int, rounding towards -Inf, clamping
.thumb_func
qfp_float2int:
movs r1,#0 @ fall through
@ convert float in r0 to signed fixed point in r0, clamping
.thumb_func
qfp_float2fix:
push {r4,r14}
bl unpackx
movs r3,r2
adds r3,#130
bmi 6f @ -0?
add r2,r1 @ incorporate binary point position into exponent
subs r2,#23 @ r2 is now amount of left shift required
blt 1f @ requires right shift?
cmp r2,#7 @ overflow?
ble 4f
3: @ overflow
asrs r1,r0,#31 @ +ve:0 -ve:0xffffffff
mvns r1,r1 @ +ve:0xffffffff -ve:0
movs r0,#1
lsls r0,#31
5:
eors r0,r1 @ +ve:0x7fffffff -ve:0x80000000 (unsigned path: 0xffffffff)
pop {r4,r15}
1:
rsbs r2,#0 @ right shift for r0, >0
cmp r2,#32
blt 2f @ more than 32 bits of right shift?
movs r2,#32
2:
asrs r0,r0,r2
pop {r4,r15}
6:
movs r0,#0
pop {r4,r15}
@ unsigned version
.thumb_func
qfp_float2uint:
movs r1,#0 @ fall through
.thumb_func
qfp_float2ufix:
push {r4,r14}
bl unpackx
add r2,r1 @ incorporate binary point position into exponent
movs r1,r0
bmi 5b @ negative? return zero
subs r2,#23 @ r2 is now amount of left shift required
blt 1b @ requires right shift?
mvns r1,r0 @ ready to return 0xffffffff
cmp r2,#8 @ overflow?
bgt 5b
4:
lsls r0,r0,r2 @ result fits, left shifted
pop {r4,r15}
@ convert uint64 to float, rounding
.thumb_func
qfp_uint642float:
movs r2,#0 @ fall through
@ convert unsigned 64-bit fix to float, rounding; number of r0:r1 bits after point in r2
.thumb_func
qfp_ufix642float:
push {r4,r5,r14}
cmp r1,#0
bpl 3f @ positive? we can use signed code
lsls r5,r1,#31 @ contribution to sticky bits
orrs r5,r0
lsrs r0,r1,#1
subs r2,#1
b 4f
@ convert int64 to float, rounding
.thumb_func
qfp_int642float:
movs r2,#0 @ fall through
@ convert signed 64-bit fix to float, rounding; number of r0:r1 bits after point in r2
.thumb_func
qfp_fix642float:
push {r4,r5,r14}
3:
movs r5,r0
orrs r5,r1
beq ret_pop45 @ zero? return +0
asrs r5,r1,#31 @ sign bits
2:
asrs r4,r1,#24 @ try shifting 7 bits at a time
cmp r4,r5
bne 1f @ next shift will overflow?
lsls r1,#7
lsrs r4,r0,#25
orrs r1,r4
lsls r0,#7
adds r2,#7
b 2b
1:
movs r5,r0
movs r0,r1
4:
rsbs r2,#0
adds r2,#32+29
b packret
@ convert signed int to float, rounding
.thumb_func
qfp_int2float:
movs r1,#0 @ fall through
@ convert signed fix to float, rounding; number of r0 bits after point in r1
.thumb_func
qfp_fix2float:
push {r4,r5,r14}
1:
movs r2,#29
subs r2,r1 @ fix exponent
packretns: @ pack and return, sticky bits=0
movs r5,#0
packret: @ common return point: "pack and return"
bl packx
ret_pop45:
pop {r4,r5,r15}
@ unsigned version
.thumb_func
qfp_uint2float:
movs r1,#0 @ fall through
.thumb_func
qfp_ufix2float:
push {r4,r5,r14}
cmp r0,#0
bge 1b @ treat <2^31 as signed
movs r2,#30
subs r2,r1 @ fix exponent
lsls r5,r0,#31 @ one sticky bit
lsrs r0,#1
b packret
@ All the scientific functions are implemented using the CORDIC algorithm. For notation,
@ details not explained in the comments below, and a good overall survey see
@ "50 Years of CORDIC: Algorithms, Architectures, and Applications" by Meher et al.,
@ IEEE Transactions on Circuits and Systems Part I, Volume 56 Issue 9.
@ Register use:
@ r0: x
@ r1: y
@ r2: z/omega
@ r3: coefficient pointer
@ r4,r12: m
@ r5: i (shift)
cordic_start: @ initialisation
movs r5,#0 @ initial shift=0
mov r12,r4
b 5f
cordic_vstep: @ one step of algorithm in vector mode
cmp r1,#0 @ check sign of y
bgt 4f
b 1f
cordic_rstep: @ one step of algorithm in rotation mode
cmp r2,#0 @ check sign of angle
bge 1f
4:
subs r1,r6 @ negative rotation: y=y-(x>>i)
rsbs r7,#0
adds r2,r4 @ accumulate angle
b 2f
1:
adds r1,r6 @ positive rotation: y=y+(x>>i)
subs r2,r4 @ accumulate angle
2:
mov r4,r12
muls r7,r4 @ apply sign from m
subs r0,r7 @ finish rotation: x=x{+/-}(y>>i)
5:
ldmia r3!,{r4} @ fetch next angle from table and bump pointer
lsrs r4,#1 @ repeated angle?
bcs 3f
adds r5,#1 @ adjust shift if not
3:
mov r6,r0
asrs r6,r5 @ x>>i
mov r7,r1
asrs r7,r5 @ y>>i
lsrs r4,#1 @ shift end flag into carry
bx r14
@ CORDIC rotation mode
cordic_rot:
push {r6,r7,r14}
bl cordic_start @ initialise
1:
bl cordic_rstep
bcc 1b @ step until table finished
asrs r6,r0,#14 @ remaining small rotations can be linearised: see IV.B of paper referenced above
asrs r7,r1,#14
asrs r2,#3
muls r6,r2 @ all remaining CORDIC steps in a multiplication
muls r7,r2
mov r4,r12
muls r7,r4
asrs r6,#12
asrs r7,#12
subs r0,r7 @ x=x{+/-}(yz>>k)
adds r1,r6 @ y=y+(xz>>k)
cordic_exit:
pop {r6,r7,r15}
@ CORDIC vector mode
cordic_vec:
push {r6,r7,r14}
bl cordic_start @ initialise
1:
bl cordic_vstep
bcc 1b @ step until table finished
4:
cmp r1,#0 @ continue as in cordic_vstep but without using table; x is not affected as y is small
bgt 2f @ check sign of y
adds r1,r6 @ positive rotation: y=y+(x>>i)
subs r2,r4 @ accumulate angle
b 3f
2:
subs r1,r6 @ negative rotation: y=y-(x>>i)
adds r2,r4 @ accumulate angle
3:
asrs r6,#1
asrs r4,#1 @ next "table entry"
bne 4b
b cordic_exit
.thumb_func
qfp_fsin: @ calculate sin and cos using CORDIC rotation method
push {r4,r5,r14}
movs r1,#24
bl qfp_float2fix @ range reduction by repeated subtraction/addition in fixed point
ldr r4,pi_q29
lsrs r4,#4 @ 2pi Q24
1:
subs r0,r4
bge 1b
1:
adds r0,r4
bmi 1b @ now in range 0..2pi
lsls r2,r0,#2 @ z Q26
lsls r5,r4,#1 @ pi Q26 (r4=pi/2 Q26)
ldr r0,=#0x136e9db4 @ initialise CORDIC x,y with scaling
movs r1,#0
1:
cmp r2,r4 @ >pi/2?
blt 2f
subs r2,r5 @ reduce range to -pi/2..pi/2
rsbs r0,#0 @ rotate vector by pi
b 1b
2:
lsls r2,#3 @ Q29
adr r3,tab_cc @ circular coefficients
movs r4,#1 @ m=1
bl cordic_rot
adds r1,#9 @ fiddle factor to make sin(0)==0
movs r2,#0 @ exponents to zero
movs r3,#0
movs r5,#0 @ no sticky bits
bl clampx
bl packx @ pack cosine
bl xchxy
bl clampx
b packretns @ pack sine
.thumb_func
qfp_fcos:
push {r14}
bl qfp_fsin
mov r0,r1 @ extract cosine result
pop {r15}
@ force r0 to lie in range [-1,1] Q29
clampx:
movs r4,#1
lsls r4,#29
cmp r0,r4
bgt 1f
rsbs r4,#0
cmp r0,r4
ble 1f
bx r14
1:
movs r0,r4
bx r14
.thumb_func
qfp_ftan:
push {r4,r5,r6,r14}
bl qfp_fsin @ sine in r0/r2, cosine in r1/r3
b fdiv_n @ sin/cos
.thumb_func
qfp_fexp:
push {r4,r5,r14}
movs r1,#24
bl qfp_float2fix @ Q24: covers entire valid input range
asrs r1,r0,#16 @ Q8
ldr r2,=#5909 @ log_2(e) Q12
muls r2,r1 @ estimate exponent of result Q20 (always an underestimate)
asrs r2,#20 @ Q0
lsls r1,r0,#6 @ Q30
ldr r0,=#0x2c5c85fe @ ln(2) Q30
muls r0,r2 @ accurate contribution of estimated exponent
subs r1,r0 @ residual to be exponentiated, guaranteed ≥0, < about 0.75 Q30
@ here
@ r1: mantissa to exponentiate, 0...~0.75 Q30
@ r2: first exponent estimate
movs r5,#1 @ shift
adr r3,ftab_exp @ could use alternate words from dtab_exp to save space if required
movs r0,#1
lsls r0,#29 @ x=1 Q29
3:
ldmia r3!,{r4}
subs r4,r1,r4
bmi 1f
movs r1,r4 @ keep result of subtraction
movs r4,r0
lsrs r4,r5
adcs r0,r4 @ x+=x>>i with rounding
1:
adds r5,#1
cmp r5,#15
bne 3b
@ here
@ r0: exp a Q29 1..2+
@ r1: ε (residual x where x=a+ε), < 2^-14 Q30
@ r2: first exponent estimate
@ and we wish to calculate exp x=exp a exp ε~(exp a)(1+ε)
lsrs r3,r0,#15 @ exp a Q14
muls r3,r1 @ ε exp a Q44
lsrs r3,#15 @ ε exp a Q29
adcs r0,r3 @ (1+ε) exp a Q29 with rounding
b packretns @ pack result
.thumb_func
qfp_fln:
push {r4,r5,r14}
asrs r1,r0,#23
bmi 3f @ -ve argument?
beq 3f @ 0 argument?
cmp r1,#0xff
beq 4f @ +Inf/NaN
bl unpackx
adds r2,#1
ldr r3,=#0x2c5c85fe @ ln(2) Q30
lsrs r1,r3,#14 @ ln(2) Q16
muls r1,r2 @ result estimate Q16
asrs r1,#16 @ integer contribution to result
muls r3,r2
lsls r4,r1,#30
subs r3,r4 @ fractional contribution to result Q30, signed
lsls r0,#8 @ Q31
@ here
@ r0: mantissa Q31
@ r1: integer contribution to result
@ r3: fractional contribution to result Q30, signed
movs r5,#1 @ shift
adr r4,ftab_exp @ could use alternate words from dtab_exp to save space if required
2:
movs r2,r0
lsrs r2,r5
adcs r2,r0 @ x+(x>>i) with rounding
bcs 1f @ >=2?
movs r0,r2 @ keep result
ldr r2,[r4]
subs r3,r2
1:
adds r4,#4
adds r5,#1
cmp r5,#15
bne 2b
@ here
@ r0: residual x, nearly 2 Q31
@ r1: integer contribution to result
@ r3: fractional part of result Q30
asrs r0,#2
adds r0,r3,r0
cmp r1,#0
bne 2f
asrs r0,#1
lsls r1,#29
adds r0,r1
movs r2,#0
b packretns
2:
lsls r1,#24
asrs r0,#6 @ Q24
adcs r0,r1 @ with rounding
movs r2,#5
b packretns
3:
ldr r0,=#0xff800000 @ -Inf
pop {r4,r5,r15}
4:
ldr r0,=#0x7f800000 @ +Inf
pop {r4,r5,r15}
.align 2
ftab_exp:
.word 0x19f323ed @ log 1+2^-1 Q30
.word 0x0e47fbe4 @ log 1+2^-2 Q30
.word 0x0789c1dc @ log 1+2^-3 Q30
.word 0x03e14618 @ log 1+2^-4 Q30
.word 0x01f829b1 @ log 1+2^-5 Q30
.word 0x00fe0546 @ log 1+2^-6 Q30
.word 0x007f80aa @ log 1+2^-7 Q30
.word 0x003fe015 @ log 1+2^-8 Q30
.word 0x001ff803 @ log 1+2^-9 Q30
.word 0x000ffe00 @ log 1+2^-10 Q30
.word 0x0007ff80 @ log 1+2^-11 Q30
.word 0x0003ffe0 @ log 1+2^-12 Q30
.word 0x0001fff8 @ log 1+2^-13 Q30
.word 0x0000fffe @ log 1+2^-14 Q30
.thumb_func
qfp_fatan2:
push {r4,r5,r14}
@ unpack arguments and shift one down to have common exponent
bl unpackx
bl xchxy
bl unpackx
lsls r0,r0,#5 @ Q28
lsls r1,r1,#5 @ Q28
adds r4,r2,r3 @ this is -760 if both arguments are 0 and at least -380-126=-506 otherwise
asrs r4,#9
adds r4,#1
bmi 2f @ force y to 0 proper, so result will be zero
subs r4,r2,r3 @ calculate shift
bge 1f @ ex>=ey?
rsbs r4,#0 @ make shift positive
asrs r0,r4
cmp r4,#28
blo 3f
asrs r0,#31
b 3f
1:
asrs r1,r4
cmp r4,#28
blo 3f
2:
@ here |x|>>|y| or both x and y are ±0
cmp r0,#0
bge 4f @ x positive, return signed 0
ldr r0,pi_q29 @ x negative, return +/- pi
asrs r1,#31
eors r0,r1
b 7f
4:
asrs r0,r1,#31
b 7f
3:
movs r2,#0 @ initial angle
cmp r0,#0 @ x negative
bge 5f
rsbs r0,#0 @ rotate to 1st/4th quadrants
rsbs r1,#0
ldr r2,pi_q29 @ pi Q29
5:
adr r3,tab_cc @ circular coefficients
movs r4,#1 @ m=1
bl cordic_vec @ also produces magnitude (with scaling factor 1.646760119), which is discarded
mov r0,r2 @ result here is -pi/2..3pi/2 Q29
@ asrs r2,#29
@ subs r0,r2
ldr r2,pi_q29 @ pi Q29
adds r4,r0,r2 @ attempt to fix -3pi/2..-pi case
bcs 6f @ -pi/2..0? leave result as is
subs r4,r0,r2 @ <pi? leave as is
bmi 6f
subs r0,r4,r2 @ >pi: take off 2pi
6:
subs r0,#1 @ fiddle factor so atan2(0,1)==0
7:
movs r2,#0 @ exponent for pack
b packretns
.align 2
.ltorg
@ first entry in following table is pi Q29
pi_q29:
@ circular CORDIC coefficients: atan(2^-i), b0=flag for preventing shift, b1=flag for end of table
tab_cc:
.word 0x1921fb54*4+1 @ no shift before first iteration
.word 0x0ed63383*4+0
.word 0x07d6dd7e*4+0
.word 0x03fab753*4+0
.word 0x01ff55bb*4+0
.word 0x00ffeaae*4+0
.word 0x007ffd55*4+0
.word 0x003fffab*4+0
.word 0x001ffff5*4+0
.word 0x000fffff*4+0
.word 0x0007ffff*4+0
.word 0x00040000*4+0
.word 0x00020000*4+0+2 @ +2 marks end
.align 2
.thumb_func
qfp_fsub:
ldr r2,=#0x80000000
eors r1,r2 @ flip sign on second argument
@ drop into fadd, on .align2:ed boundary
.thumb_func
qfp_fadd:
push {r4,r5,r6,r14}
asrs r4,r0,#31
lsls r2,r0,#1
lsrs r2,#24 @ x exponent
beq fa_xe0
cmp r2,#255
beq fa_xe255
fa_xe:
asrs r5,r1,#31
lsls r3,r1,#1
lsrs r3,#24 @ y exponent
beq fa_ye0
cmp r3,#255
beq fa_ye255
fa_ye:
ldr r6,=#0x007fffff
ands r0,r0,r6 @ extract mantissa bits
ands r1,r1,r6
adds r6,#1 @ r6=0x00800000
orrs r0,r0,r6 @ set implied 1
orrs r1,r1,r6
eors r0,r0,r4 @ complement...
eors r1,r1,r5
subs r0,r0,r4 @ ... and add 1 if sign bit is set: 2's complement
subs r1,r1,r5
subs r5,r3,r2 @ ye-xe
subs r4,r2,r3 @ xe-ye
bmi fa_ygtx
@ here xe>=ye
cmp r4,#30
bge fa_xmgty @ xe much greater than ye?
adds r5,#32
movs r3,r2 @ save exponent
@ here y in r1 must be shifted down r4 places to align with x in r0
movs r2,r1
lsls r2,r2,r5 @ keep the bits we will shift off the bottom of r1
asrs r1,r1,r4
b fa_0
.ltorg
fa_ymgtx:
movs r2,#0 @ result is just y
movs r0,r1
b fa_1
fa_xmgty:
movs r3,r2 @ result is just x
movs r2,#0
b fa_1
fa_ygtx:
@ here ye>xe
cmp r5,#30
bge fa_ymgtx @ ye much greater than xe?
adds r4,#32
@ here x in r0 must be shifted down r5 places to align with y in r1
movs r2,r0
lsls r2,r2,r4 @ keep the bits we will shift off the bottom of r0
asrs r0,r0,r5
fa_0:
adds r0,r1 @ result is now in r0:r2, possibly highly denormalised or zero; exponent in r3
beq fa_9 @ if zero, inputs must have been of identical magnitude and opposite sign, so return +0
fa_1:
lsrs r1,r0,#31 @ sign bit
beq fa_8
mvns r0,r0
rsbs r2,r2,#0
bne fa_8
adds r0,#1
fa_8:
adds r6,r6
@ r6=0x01000000
cmp r0,r6
bhs fa_2
fa_3:
adds r2,r2 @ normalisation loop
adcs r0,r0
subs r3,#1 @ adjust exponent
cmp r0,r6
blo fa_3
fa_2:
@ here r0:r2 is the result mantissa 0x01000000<=r0<0x02000000, r3 the exponent, and r1 the sign bit
lsrs r0,#1
bcc fa_4
@ rounding bits here are 1:r2
adds r0,#1 @ round up
cmp r2,#0
beq fa_5 @ sticky bits all zero?
fa_4:
cmp r3,#254
bhs fa_6 @ exponent too large or negative?
lsls r1,#31 @ pack everything
add r0,r1
lsls r3,#23
add r0,r3
fa_end:
pop {r4,r5,r6,r15}
fa_9:
cmp r2,#0 @ result zero?
beq fa_end @ return +0
b fa_1
fa_5:
lsrs r0,#1
lsls r0,#1 @ round to even
b fa_4
fa_6:
bge fa_7
@ underflow
@ can handle denormals here
lsls r0,r1,#31 @ result is signed zero
pop {r4,r5,r6,r15}
fa_7:
@ overflow
lsls r0,r1,#8
adds r0,#255
lsls r0,#23 @ result is signed infinity
pop {r4,r5,r6,r15}
fa_xe0:
@ can handle denormals here
subs r2,#32
adds r2,r4 @ exponent -32 for +Inf, -33 for -Inf
b fa_xe
fa_xe255:
@ can handle NaNs here
lsls r2,#8
add r2,r2,r4 @ exponent ~64k for +Inf, ~64k-1 for -Inf
b fa_xe
fa_ye0:
@ can handle denormals here
subs r3,#32
adds r3,r5 @ exponent -32 for +Inf, -33 for -Inf
b fa_ye
fa_ye255:
@ can handle NaNs here
lsls r3,#8
add r3,r3,r5 @ exponent ~64k for +Inf, ~64k-1 for -Inf
b fa_ye
.align 2
.thumb_func
qfp_fmul:
push {r7,r14}
mov r2,r0
eors r2,r1 @ sign of result
lsrs r2,#31
lsls r2,#31
mov r14,r2
lsls r0,#1
lsls r1,#1
lsrs r2,r0,#24 @ xe
beq fm_xe0
cmp r2,#255
beq fm_xe255
fm_xe:
lsrs r3,r1,#24 @ ye
beq fm_ye0
cmp r3,#255
beq fm_ye255
fm_ye:
adds r7,r2,r3 @ exponent of result (will possibly be incremented)
subs r7,#128 @ adjust bias for packing
lsls r0,#8 @ x mantissa
lsls r1,#8 @ y mantissa
lsrs r0,#9
lsrs r1,#9
adds r2,r0,r1 @ for later
mov r12,r2
lsrs r2,r0,#7 @ x[22..7] Q16
lsrs r3,r1,#7 @ y[22..7] Q16
muls r2,r2,r3 @ result [45..14] Q32: never an overestimate and worst case error is 2*(2^7-1)*(2^23-2^7)+(2^7-1)^2 = 2130690049 < 2^31
muls r0,r0,r1 @ result [31..0] Q46
lsrs r2,#18 @ result [45..32] Q14
bcc 1f
cmp r0,#0
bmi 1f
adds r2,#1 @ fix error in r2
1:
lsls r3,r0,#9 @ bits off bottom of result
lsrs r0,#23 @ Q23
lsls r2,#9
adds r0,r2 @ cut'n'shut
add r0,r12 @ implied 1*(x+y) to compensate for no insertion of implied 1s
@ result-1 in r3:r0 Q23+32, i.e., in range [0,3)
lsrs r1,r0,#23
bne fm_0 @ branch if we need to shift down one place
@ here 1<=result<2
cmp r7,#254
bhs fm_3a @ catches both underflow and overflow
lsls r3,#1 @ sticky bits at top of R3, rounding bit in carry
bcc fm_1 @ no rounding
beq fm_2 @ rounding tie?
adds r0,#1 @ round up
fm_1:
adds r7,#1 @ for implied 1
lsls r7,#23 @ pack result
add r0,r7
add r0,r14
pop {r7,r15}
fm_2: @ rounding tie
adds r0,#1
fm_3:
lsrs r0,#1
lsls r0,#1 @ clear bottom bit
b fm_1
@ here 1<=result-1<3
fm_0:
adds r7,#1 @ increment exponent
cmp r7,#254
bhs fm_3b @ catches both underflow and overflow
lsrs r0,#1 @ shift mantissa down
bcc fm_1a @ no rounding
adds r0,#1 @ assume we will round up
cmp r3,#0 @ sticky bits
beq fm_3c @ rounding tie?
fm_1a:
adds r7,r7
adds r7,#1 @ for implied 1
lsls r7,#22 @ pack result
add r0,r7
add r0,r14
pop {r7,r15}
fm_3c:
lsrs r0,#1
lsls r0,#1 @ clear bottom bit
b fm_1a
fm_xe0:
subs r2,#16
fm_xe255:
lsls r2,#8
b fm_xe
fm_ye0:
subs r3,#16
fm_ye255:
lsls r3,#8
b fm_ye
@ here the result is under- or overflowing
fm_3b:
bge fm_4 @ branch on overflow
@ trap case where result is denormal 0x007fffff + 0.5ulp or more
adds r7,#1 @ exponent=-1?
bne fm_5
@ corrected mantissa will be >= 3.FFFFFC (0x1fffffe Q23)
@ so r0 >= 2.FFFFFC (0x17ffffe Q23)
adds r0,#2
lsrs r0,#23
cmp r0,#3
bne fm_5
b fm_6
fm_3a:
bge fm_4 @ branch on overflow
@ trap case where result is denormal 0x007fffff + 0.5ulp or more
adds r7,#1 @ exponent=-1?
bne fm_5
adds r0,#1 @ mantissa=0xffffff (i.e., r0=0x7fffff)?
lsrs r0,#23
beq fm_5
fm_6:
movs r0,#1 @ return smallest normal
lsls r0,#23
add r0,r14
pop {r7,r15}
fm_5:
mov r0,r14
pop {r7,r15}
fm_4:
movs r0,#0xff
lsls r0,#23
add r0,r14
pop {r7,r15}
@ This version of the division algorithm uses external divider hardware to estimate the
@ reciprocal of the divisor to about 14 bits; then a multiplication step to get a first
@ quotient estimate; then the remainder based on this estimate is used to calculate a
@ correction to the quotient. The result is good to about 27 bits and so we only need
@ to calculate the exact remainder when close to a rounding boundary.
.align 2
.thumb_func
qfp_fdiv:
push {r4,r5,r6,r14}
fdiv_n:
movs r4,#1
lsls r4,#23 @ implied 1 position
lsls r2,r1,#9 @ clear out sign and exponent
lsrs r2,r2,#9
orrs r2,r2,r4 @ divisor mantissa Q23 with implied 1
@ here
@ r0=packed dividend
@ r1=packed divisor
@ r2=divisor mantissa Q23
@ r4=1<<23
// see divtest.c
lsrs r3,r2,#18 @ x2=x>>18; // Q5 32..63
adr r5,rcpapp-32
ldrb r3,[r5,r3] @ u=lut5[x2-32]; // Q8
lsls r5,r2,#5
muls r5,r5,r3
asrs r5,#14 @ e=(i32)(u*(x<<5))>>14; // Q22
asrs r6,r5,#11
muls r6,r6,r6 @ e2=(e>>11)*(e>>11); // Q22
subs r5,r6
muls r5,r5,r3 @ c=(e-e2)*u; // Q30
lsls r6,r3,#8
asrs r5,#13
adds r5,#1
asrs r5,#1
subs r5,r6,r5 @ u0=(u<<8)-((c+0x2000)>>14); // Q16
@ here
@ r0=packed dividend
@ r1=packed divisor
@ r2=divisor mantissa Q23
@ r4=1<<23
@ r5=reciprocal estimate Q16
lsrs r6,r0,#23
uxtb r3,r6 @ dividend exponent
lsls r0,#9
lsrs r0,#9
orrs r0,r0,r4 @ dividend mantissa Q23
lsrs r1,#23
eors r6,r1 @ sign of result in bit 8
lsrs r6,#8
lsls r6,#31 @ sign of result in bit 31, other bits clear
@ here
@ r0=dividend mantissa Q23
@ r1=divisor sign+exponent
@ r2=divisor mantissa Q23
@ r3=dividend exponent
@ r5=reciprocal estimate Q16
@ r6b31=sign of result
uxtb r1,r1 @ divisor exponent
cmp r1,#0
beq retinf
cmp r1,#255
beq 20f @ divisor is infinite
cmp r3,#0
beq retzero
cmp r3,#255
beq retinf
subs r3,r1 @ initial result exponent (no bias)
adds r3,#125 @ add bias
lsrs r1,r0,#8 @ dividend mantissa Q15
@ here
@ r0=dividend mantissa Q23
@ r1=dividend mantissa Q15
@ r2=divisor mantissa Q23
@ r3=initial result exponent
@ r5=reciprocal estimate Q16
@ r6b31=sign of result
muls r1,r5
lsrs r1,#16 @ Q15 qu0=(q15)(u*y0);
lsls r0,r0,#15 @ dividend Q38
movs r4,r2
muls r4,r1 @ Q38 qu0*x
subs r4,r0,r4 @ Q38 re0=(y<<15)-qu0*x; note this remainder is signed
asrs r4,#10
muls r4,r5 @ Q44 qu1=(re0>>10)*u; this quotient correction is also signed
asrs r4,#16 @ Q28
lsls r1,#13
adds r1,r1,r4 @ Q28 qu=(qu0<<13)+(qu1>>16);
@ here
@ r0=dividend mantissa Q38
@ r1=quotient Q28
@ r2=divisor mantissa Q23
@ r3=initial result exponent
@ r6b31=sign of result
lsrs r4,r1,#28
bne 1f
@ here the quotient is less than 1<<28 (i.e., result mantissa <1.0)
adds r1,#5
lsrs r4,r1,#4 @ rounding + small reduction in systematic bias
bcc 2f @ skip if we are not near a rounding boundary
lsrs r1,#3 @ quotient Q25
lsls r0,#10 @ dividend mantissa Q48
muls r1,r1,r2 @ quotient*divisor Q48
subs r0,r0,r1 @ remainder Q48
bmi 2f
b 3f
1:
@ here the quotient is at least 1<<28 (i.e., result mantissa >=1.0)
adds r3,#1 @ bump exponent (and shift mantissa down one more place)
adds r1,#9
lsrs r4,r1,#5 @ rounding + small reduction in systematic bias
bcc 2f @ skip if we are not near a rounding boundary
lsrs r1,#4 @ quotient Q24
lsls r0,#9 @ dividend mantissa Q47
muls r1,r1,r2 @ quotient*divisor Q47
subs r0,r0,r1 @ remainder Q47
bmi 2f
3:
adds r4,#1 @ increment quotient as we are above the rounding boundary
@ here
@ r3=result exponent
@ r4=correctly rounded quotient Q23 in range [1,2] *note closed interval*
@ r6b31=sign of result
2:
cmp r3,#254
bhs 10f @ this catches both underflow and overflow
lsls r1,r3,#23
adds r0,r4,r1
adds r0,r6
pop {r4,r5,r6,r15}
@ here divisor is infinite; dividend exponent in r3
20:
cmp r3,#255
bne retzero
retinf:
movs r0,#255
21:
lsls r0,#23
orrs r0,r6
pop {r4,r5,r6,r15}
10:
bge retinf @ overflow?
adds r1,r3,#1
bne retzero @ exponent <-1? return 0
@ here exponent is exactly -1
lsrs r1,r4,#25
bcc retzero @ mantissa is not 01000000?
@ return minimum normal
movs r0,#1
lsls r0,#23
orrs r0,r6
pop {r4,r5,r6,r15}
retzero:
movs r0,r6
pop {r4,r5,r6,r15}
@ x2=[32:1:63]/32;
@ round(256 ./(x2+1/64))
.align 2
rcpapp:
.byte 252,245,237,231,224,218,213,207,202,197,193,188,184,180,176,172
.byte 169,165,162,159,156,153,150,148,145,142,140,138,135,133,131,129
@ The square root routine uses an initial approximation to the reciprocal of the square root of the argument based
@ on the top four bits of the mantissa (possibly shifted one place to make the exponent even). It then performs two
@ Newton-Raphson iterations, resulting in about 14 bits of accuracy. This reciprocal is then multiplied by
@ the original argument to produce an approximation to the result, again with about 14 bits of accuracy.
@ Then a remainder is calculated, and multiplied by the reciprocal estiamte to generate a correction term
@ giving a final answer to about 28 bits of accuracy. A final remainder calculation rounds to the correct
@ result if necessary.
@ Again, the fixed-point calculation is carefully implemented to preserve accuracy, and similar comments to those
@ made above on the fast division routine apply.
@ The reciprocal square root calculation has been tested for all possible (possibly shifted) input mantissa values.
.align 2
.thumb_func
qfp_fsqrt:
push {r4}
lsls r1,r0,#1
bcs sq_0 @ negative?
lsls r1,#8
lsrs r1,#9 @ mantissa
movs r2,#1
lsls r2,#23
adds r1,r2 @ insert implied 1
lsrs r2,r0,#23 @ extract exponent
beq sq_2 @ zero?
cmp r2,#255 @ infinite?
beq sq_1
adds r2,#125 @ correction for packing
asrs r2,#1 @ exponent/2, LSB into carry
bcc 1f
lsls r1,#1 @ was even: double mantissa; mantissa y now 1..4 Q23
1:
adr r4,rsqrtapp-4@ first four table entries are never accessed because of the mantissa's leading 1
lsrs r3,r1,#21 @ y Q2
ldrb r4,[r4,r3] @ initial approximation to reciprocal square root a0 Q8
lsrs r0,r1,#7 @ y Q16: first Newton-Raphson iteration
muls r0,r4 @ a0*y Q24
muls r0,r4 @ r0=p0=a0*y*y Q32
asrs r0,#12 @ r0 Q20
muls r0,r4 @ dy0=a0*r0 Q28
asrs r0,#13 @ dy0 Q15
lsls r4,#8 @ a0 Q16
subs r4,r0 @ a1=a0-dy0/2 Q16-Q15/2 -> Q16
adds r4,#170 @ mostly remove systematic error in this approximation: gains approximately 1 bit
movs r0,r4 @ second Newton-Raphson iteration
muls r0,r0 @ a1*a1 Q32
lsrs r0,#15 @ a1*a1 Q17
lsrs r3,r1,#8 @ y Q15
muls r0,r3 @ r1=p1=a1*a1*y Q32
asrs r0,#12 @ r1 Q20
muls r0,r4 @ dy1=a1*r1 Q36
asrs r0,#21 @ dy1 Q15
subs r4,r0 @ a2=a1-dy1/2 Q16-Q15/2 -> Q16
muls r3,r4 @ a3=y*a2 Q31
lsrs r3,#15 @ a3 Q16
@ here a2 is an approximation to the reciprocal square root
@ and a3 is an approximation to the square root
movs r0,r3
muls r0,r0 @ a3*a3 Q32
lsls r1,#9 @ y Q32
subs r0,r1,r0 @ r2=y-a3*a3 Q32 remainder
asrs r0,#5 @ r2 Q27
muls r4,r0 @ r2*a2 Q43
lsls r3,#7 @ a3 Q23
asrs r0,r4,#15 @ r2*a2 Q28
adds r0,#16 @ rounding to Q24
asrs r0,r0,#6 @ r2*a2 Q22
add r3,r0 @ a4 Q23: candidate final result
bcc sq_3 @ near rounding boundary? skip if no rounding needed
mov r4,r3
adcs r4,r4 @ a4+0.5ulp Q24
muls r4,r4 @ Q48
lsls r1,#16 @ y Q48
subs r1,r4 @ remainder Q48
bmi sq_3
adds r3,#1 @ round up
sq_3:
lsls r2,#23 @ pack exponent
adds r0,r2,r3
sq_6:
pop {r4}
bx r14
sq_0:
lsrs r1,#24
beq sq_2 @ -0: return it
@ here negative and not -0: return -Inf
asrs r0,#31
sq_5:
lsls r0,#23
b sq_6
sq_1: @ +Inf
lsrs r0,#23
b sq_5
sq_2:
lsrs r0,#31
lsls r0,#31
b sq_6
@ round(sqrt(2^22./[72:16:248]))
rsqrtapp:
.byte 0xf1,0xda,0xc9,0xbb, 0xb0,0xa6,0x9e,0x97, 0x91,0x8b,0x86,0x82
@ Notation:
@ rx:ry means the concatenation of rx and ry with rx having the less significant bits
@ IEEE double in ra:rb ->
@ mantissa in ra:rb 12Q52 (53 significant bits) with implied 1 set
@ exponent in re
@ sign in rs
@ trashes rt
.macro mdunpack ra,rb,re,rs,rt
lsrs \re,\rb,#20 @ extract sign and exponent
subs \rs,\re,#1
lsls \rs,#20
subs \rb,\rs @ clear sign and exponent in mantissa; insert implied 1
lsrs \rs,\re,#11 @ sign
lsls \re,#21
lsrs \re,#21 @ exponent
beq l\@_1 @ zero exponent?
adds \rt,\re,#1
lsrs \rt,#11
beq l\@_2 @ exponent != 0x7ff? then done
l\@_1:
movs \ra,#0
movs \rb,#1
lsls \rb,#20
subs \re,#128
lsls \re,#12
l\@_2:
.endm
@ IEEE double in ra:rb ->
@ signed mantissa in ra:rb 12Q52 (53 significant bits) with implied 1
@ exponent in re
@ trashes rt0 and rt1
@ +zero, +denormal -> exponent=-0x80000
@ -zero, -denormal -> exponent=-0x80000
@ +Inf, +NaN -> exponent=+0x77f000
@ -Inf, -NaN -> exponent=+0x77e000
.macro mdunpacks ra,rb,re,rt0,rt1
lsrs \re,\rb,#20 @ extract sign and exponent
lsrs \rt1,\rb,#31 @ sign only
subs \rt0,\re,#1
lsls \rt0,#20
subs \rb,\rt0 @ clear sign and exponent in mantissa; insert implied 1
lsls \re,#21
bcc l\@_1 @ skip on positive
mvns \rb,\rb @ negate mantissa
rsbs \ra,#0
bcc l\@_1
adds \rb,#1
l\@_1:
lsrs \re,#21
beq l\@_2 @ zero exponent?
adds \rt0,\re,#1
lsrs \rt0,#11
beq l\@_3 @ exponent != 0x7ff? then done
subs \re,\rt1
l\@_2:
movs \ra,#0
lsls \rt1,#1 @ +ve: 0 -ve: 2
adds \rb,\rt1,#1 @ +ve: 1 -ve: 3
lsls \rb,#30 @ create +/-1 mantissa
asrs \rb,#10
subs \re,#128
lsls \re,#12
l\@_3:
.endm
.align 2
.thumb_func
qfp_dsub:
push {r4-r7,r14}
movs r4,#1
lsls r4,#31
eors r3,r4 @ flip sign on second argument
b da_entry @ continue in dadd
.align 2
.thumb_func
qfp_dadd:
push {r4-r7,r14}
da_entry:
mdunpacks r0,r1,r4,r6,r7
mdunpacks r2,r3,r5,r6,r7
subs r7,r5,r4 @ ye-xe
subs r6,r4,r5 @ xe-ye
bmi da_ygtx
@ here xe>=ye: need to shift y down r6 places
mov r12,r4 @ save exponent
cmp r6,#32
bge da_xrgty @ xe rather greater than ye?
adds r7,#32
movs r4,r2
lsls r4,r4,r7 @ rounding bit + sticky bits
da_xgty0:
movs r5,r3
lsls r5,r5,r7
lsrs r2,r6
asrs r3,r6
orrs r2,r5
da_add:
adds r0,r2
adcs r1,r3
da_pack:
@ here unnormalised signed result (possibly 0) is in r0:r1 with exponent r12, rounding + sticky bits in r4
@ Note that if a large normalisation shift is required then the arguments were close in magnitude and so we
@ cannot have not gone via the xrgty/yrgtx paths. There will therefore always be enough high bits in r4
@ to provide a correct continuation of the exact result.
@ now pack result back up
lsrs r3,r1,#31 @ get sign bit
beq 1f @ skip on positive
mvns r1,r1 @ negate mantissa
mvns r0,r0
movs r2,#0
rsbs r4,#0
adcs r0,r2
adcs r1,r2
1:
mov r2,r12 @ get exponent
lsrs r5,r1,#21
bne da_0 @ shift down required?
lsrs r5,r1,#20
bne da_1 @ normalised?
cmp r0,#0
beq da_5 @ could mantissa be zero?
da_2:
adds r4,r4
adcs r0,r0
adcs r1,r1
subs r2,#1 @ adjust exponent
lsrs r5,r1,#20
beq da_2
da_1:
lsls r4,#1 @ check rounding bit
bcc da_3
da_4:
adds r0,#1 @ round up
bcc 2f
adds r1,#1
2:
cmp r4,#0 @ sticky bits zero?
bne da_3
lsrs r0,#1 @ round to even
lsls r0,#1
da_3:
subs r2,#1
bmi da_6
adds r4,r2,#2 @ check if exponent is overflowing
lsrs r4,#11
bne da_7
lsls r2,#20 @ pack exponent and sign
add r1,r2
lsls r3,#31
add r1,r3
pop {r4-r7,r15}
da_7:
@ here exponent overflow: return signed infinity
lsls r1,r3,#31
ldr r3,=#0x7ff00000
orrs r1,r3
b 1f
da_6:
@ here exponent underflow: return signed zero
lsls r1,r3,#31
1:
movs r0,#0
pop {r4-r7,r15}
da_5:
@ here mantissa could be zero
cmp r1,#0
bne da_2
cmp r4,#0
bne da_2
@ inputs must have been of identical magnitude and opposite sign, so return +0
pop {r4-r7,r15}
da_0:
@ here a shift down by one place is required for normalisation
adds r2,#1 @ adjust exponent
lsls r6,r0,#31 @ save rounding bit
lsrs r0,#1
lsls r5,r1,#31
orrs r0,r5
lsrs r1,#1
cmp r6,#0
beq da_3
b da_4
da_xrgty: @ xe>ye and shift>=32 places
cmp r6,#60
bge da_xmgty @ xe much greater than ye?
subs r6,#32
adds r7,#64
movs r4,r2
lsls r4,r4,r7 @ these would be shifted off the bottom of the sticky bits
beq 1f
movs r4,#1
1:
lsrs r2,r2,r6
orrs r4,r2
movs r2,r3
lsls r3,r3,r7
orrs r4,r3
asrs r3,r2,#31 @ propagate sign bit
b da_xgty0
da_ygtx:
@ here ye>xe: need to shift x down r7 places
mov r12,r5 @ save exponent
cmp r7,#32
bge da_yrgtx @ ye rather greater than xe?
adds r6,#32
movs r4,r0
lsls r4,r4,r6 @ rounding bit + sticky bits
da_ygtx0:
movs r5,r1
lsls r5,r5,r6
lsrs r0,r7
asrs r1,r7
orrs r0,r5
b da_add
da_yrgtx:
cmp r7,#60
bge da_ymgtx @ ye much greater than xe?
subs r7,#32
adds r6,#64
movs r4,r0
lsls r4,r4,r6 @ these would be shifted off the bottom of the sticky bits
beq 1f
movs r4,#1
1:
lsrs r0,r0,r7
orrs r4,r0
movs r0,r1
lsls r1,r1,r6
orrs r4,r1
asrs r1,r0,#31 @ propagate sign bit
b da_ygtx0
da_ymgtx: @ result is just y
movs r0,r2
movs r1,r3
da_xmgty: @ result is just x
movs r4,#0 @ clear sticky bits
b da_pack
.ltorg
@ equivalent of UMULL
@ needs five temporary registers
@ can have rt3==rx, in which case rx trashed
@ can have rt4==ry, in which case ry trashed
@ can have rzl==rx
@ can have rzh==ry
@ can have rzl,rzh==rt3,rt4
.macro mul32_32_64 rx,ry,rzl,rzh,rt0,rt1,rt2,rt3,rt4
@ t0 t1 t2 t3 t4
@ (x) (y)
uxth \rt0,\rx @ xl
uxth \rt1,\ry @ yl
muls \rt0,\rt1 @ xlyl=L
lsrs \rt2,\rx,#16 @ xh
muls \rt1,\rt2 @ xhyl=M0
lsrs \rt4,\ry,#16 @ yh
muls \rt2,\rt4 @ xhyh=H
uxth \rt3,\rx @ xl
muls \rt3,\rt4 @ xlyh=M1
adds \rt1,\rt3 @ M0+M1=M
bcc l\@_1 @ addition of the two cross terms can overflow, so add carry into H
movs \rt3,#1 @ 1
lsls \rt3,#16 @ 0x10000
adds \rt2,\rt3 @ H'
l\@_1:
@ t0 t1 t2 t3 t4
@ (zl) (zh)
lsls \rzl,\rt1,#16 @ ML
lsrs \rzh,\rt1,#16 @ MH
adds \rzl,\rt0 @ ZL
adcs \rzh,\rt2 @ ZH
.endm
@ SUMULL: x signed, y unsigned
@ in table below ¯ means signed variable
@ needs five temporary registers
@ can have rt3==rx, in which case rx trashed
@ can have rt4==ry, in which case ry trashed
@ can have rzl==rx
@ can have rzh==ry
@ can have rzl,rzh==rt3,rt4
.macro muls32_32_64 rx,ry,rzl,rzh,rt0,rt1,rt2,rt3,rt4
@ t0 t1 t2 t3 t4
@ ¯(x) (y)
uxth \rt0,\rx @ xl
uxth \rt1,\ry @ yl
muls \rt0,\rt1 @ xlyl=L
asrs \rt2,\rx,#16 @ ¯xh
muls \rt1,\rt2 @ ¯xhyl=M0
lsrs \rt4,\ry,#16 @ yh
muls \rt2,\rt4 @ ¯xhyh=H
uxth \rt3,\rx @ xl
muls \rt3,\rt4 @ xlyh=M1
asrs \rt4,\rt1,#31 @ M0sx (M1 sign extension is zero)
adds \rt1,\rt3 @ M0+M1=M
movs \rt3,#0 @ 0
adcs \rt4,\rt3 @ ¯Msx
lsls \rt4,#16 @ ¯Msx<<16
adds \rt2,\rt4 @ H'
@ t0 t1 t2 t3 t4
@ (zl) (zh)
lsls \rzl,\rt1,#16 @ M~
lsrs \rzh,\rt1,#16 @ M~
adds \rzl,\rt0 @ ZL
adcs \rzh,\rt2 @ ¯ZH
.endm
@ SSMULL: x signed, y signed
@ in table below ¯ means signed variable
@ needs five temporary registers
@ can have rt3==rx, in which case rx trashed
@ can have rt4==ry, in which case ry trashed
@ can have rzl==rx
@ can have rzh==ry
@ can have rzl,rzh==rt3,rt4
.macro muls32_s32_64 rx,ry,rzl,rzh,rt0,rt1,rt2,rt3,rt4
@ t0 t1 t2 t3 t4
@ ¯(x) (y)
uxth \rt0,\rx @ xl
uxth \rt1,\ry @ yl
muls \rt0,\rt1 @ xlyl=L
asrs \rt2,\rx,#16 @ ¯xh
muls \rt1,\rt2 @ ¯xhyl=M0
asrs \rt4,\ry,#16 @ ¯yh
muls \rt2,\rt4 @ ¯xhyh=H
uxth \rt3,\rx @ xl
muls \rt3,\rt4 @ ¯xlyh=M1
adds \rt1,\rt3 @ ¯M0+M1=M
asrs \rt3,\rt1,#31 @ Msx
bvc l\@_1 @
mvns \rt3,\rt3 @ ¯Msx flip sign extension bits if overflow
l\@_1:
lsls \rt3,#16 @ ¯Msx<<16
adds \rt2,\rt3 @ H'
@ t0 t1 t2 t3 t4
@ (zl) (zh)
lsls \rzl,\rt1,#16 @ M~
lsrs \rzh,\rt1,#16 @ M~
adds \rzl,\rt0 @ ZL
adcs \rzh,\rt2 @ ¯ZH
.endm
@ can have rt2==rx, in which case rx trashed
@ can have rzl==rx
@ can have rzh==rt1
.macro square32_64 rx,rzl,rzh,rt0,rt1,rt2
@ t0 t1 t2 zl zh
uxth \rt0,\rx @ xl
muls \rt0,\rt0 @ xlxl=L
uxth \rt1,\rx @ xl
lsrs \rt2,\rx,#16 @ xh
muls \rt1,\rt2 @ xlxh=M
muls \rt2,\rt2 @ xhxh=H
lsls \rzl,\rt1,#17 @ ML
lsrs \rzh,\rt1,#15 @ MH
adds \rzl,\rt0 @ ZL
adcs \rzh,\rt2 @ ZH
.endm
.align 2
.thumb_func
qfp_dmul:
push {r4-r7,r14}
mdunpack r0,r1,r4,r6,r5
mov r12,r4
mdunpack r2,r3,r4,r7,r5
eors r7,r6 @ sign of result
add r4,r12 @ exponent of result
push {r0-r2,r4,r7}
@ accumulate full product in r12:r5:r6:r7
mul32_32_64 r0,r2, r0,r5, r4,r6,r7,r0,r5 @ XL*YL
mov r12,r0 @ save LL bits
mul32_32_64 r1,r3, r6,r7, r0,r2,r4,r6,r7 @ XH*YH
pop {r0} @ XL
mul32_32_64 r0,r3, r0,r3, r1,r2,r4,r0,r3 @ XL*YH
adds r5,r0
adcs r6,r3
movs r0,#0
adcs r7,r0
pop {r1,r2} @ XH,YL
mul32_32_64 r1,r2, r1,r2, r0,r3,r4, r1,r2 @ XH*YL
adds r5,r1
adcs r6,r2
movs r0,#0
adcs r7,r0
@ here r5:r6:r7 holds the product [1..4) in Q(104-32)=Q72, with extra LSBs in r12
pop {r3,r4} @ exponent in r3, sign in r4
lsls r1,r7,#11
lsrs r2,r6,#21
orrs r1,r2
lsls r0,r6,#11
lsrs r2,r5,#21
orrs r0,r2
lsls r5,#11 @ now r5:r0:r1 Q83=Q(51+32), extra LSBs in r12
lsrs r2,r1,#20
bne 1f @ skip if in range [2..4)
adds r5,r5 @ shift up so always [2..4) Q83, i.e. [1..2) Q84=Q(52+32)
adcs r0,r0
adcs r1,r1
subs r3,#1 @ correct exponent
1:
ldr r6,=#0x3ff
subs r3,r6 @ correct for exponent bias
lsls r6,#1 @ 0x7fe
cmp r3,r6
bhs dm_0 @ exponent over- or underflow
lsls r5,#1 @ rounding bit to carry
bcc 1f @ result is correctly rounded
adds r0,#1
movs r6,#0
adcs r1,r6 @ round up
mov r6,r12 @ remaining sticky bits
orrs r5,r6
bne 1f @ some sticky bits set?
lsrs r0,#1
lsls r0,#1 @ round to even
1:
lsls r3,#20
adds r1,r3
dm_2:
lsls r4,#31
add r1,r4
pop {r4-r7,r15}
@ here for exponent over- or underflow
dm_0:
bge dm_1 @ overflow?
adds r3,#1 @ would-be zero exponent?
bne 1f
adds r0,#1
bne 1f @ all-ones mantissa?
adds r1,#1
lsrs r7,r1,#21
beq 1f
lsrs r1,#1
b dm_2
1:
lsls r1,r4,#31
movs r0,#0
pop {r4-r7,r15}
@ here for exponent overflow
dm_1:
adds r6,#1 @ 0x7ff
lsls r1,r6,#20
movs r0,#0
b dm_2
.ltorg
@ Approach to division y/x is as follows.
@
@ First generate u1, an approximation to 1/x to about 29 bits. Multiply this by the top
@ 32 bits of y to generate a0, a first approximation to the result (good to 28 bits or so).
@ Calculate the exact remainder r0=y-a0*x, which will be about 0. Calculate a correction
@ d0=r0*u1, and then write a1=a0+d0. If near a rounding boundary, compute the exact
@ remainder r1=y-a1*x (which can be done using r0 as a basis) to determine whether to
@ round up or down.
@
@ The calculation of 1/x is as given in dreciptest.c. That code verifies exhaustively
@ that | u1*x-1 | < 10*2^-32.
@
@ More precisely:
@
@ x0=(q16)x;
@ x1=(q30)x;
@ y0=(q31)y;
@ u0=(q15~)"(0xffffffffU/(unsigned int)roundq(x/x_ulp))/powq(2,16)"(x0); // q15 approximation to 1/x; "~" denotes rounding rather than truncation
@ v=(q30)(u0*x1-1);
@ u1=(q30)u0-(q30~)(u0*v);
@
@ a0=(q30)(u1*y0);
@ r0=(q82)y-a0*x;
@ r0x=(q57)r0;
@ d0=r0x*u1;
@ a1=d0+a0;
@
@ Error analysis
@
@ Use Greek letters to represent the errors introduced by rounding and truncation.
@
@ r₀ = y - a₀x
@ = y - [ u₁ ( y - α ) - β ] x where 0 ≤ α < 2^-31, 0 ≤ β < 2^-30
@ = y ( 1 - u₁x ) + ( u₁α + β ) x
@
@ Hence
@
@ | r₀ / x | < 2 * 10*2^-32 + 2^-31 + 2^-30
@ = 26*2^-32
@
@ r₁ = y - a₁x
@ = y - a₀x - d₀x
@ = r₀ - d₀x
@ = r₀ - u₁ ( r₀ - γ ) x where 0 ≤ γ < 2^-57
@ = r₀ ( 1 - u₁x ) + u₁γx
@
@ Hence
@
@ | r₁ / x | < 26*2^-32 * 10*2^-32 + 2^-57
@ = (260+128)*2^-64
@ < 2^-55
@
@ Empirically it seems to be nearly twice as good as this.
@
@ To determine correctly whether the exact remainder calculation can be skipped we need a result
@ accurate to < 0.25ulp. In the case where x>y the quotient will be shifted up one place for normalisation
@ and so 1ulp is 2^-53 and so the calculation above suffices.
.align 2
.thumb_func
qfp_ddiv:
push {r4-r7,r14}
ddiv0: @ entry point from dtan
mdunpack r2,r3,r4,r7,r6 @ unpack divisor
@ unpack dividend by hand to save on register use
lsrs r6,r1,#31
adds r6,r7
mov r12,r6 @ result sign in r12b0; r12b1 trashed
lsls r1,#1
lsrs r7,r1,#21 @ exponent
beq 1f @ zero exponent?
adds r6,r7,#1
lsrs r6,#11
beq 2f @ exponent != 0x7ff? then done
1:
movs r0,#0
movs r1,#0
subs r7,#64 @ less drastic fiddling of exponents to get 0/0, Inf/Inf correct
lsls r7,#12
2:
subs r6,r7,r4
lsls r6,#2
add r12,r12,r6 @ (signed) exponent in r12[31..8]
subs r7,#1 @ implied 1
lsls r7,#21
subs r1,r7
lsrs r1,#1
// see dreciptest-boxc.c
lsrs r4,r3,#15 @ x2=x>>15; // Q5 32..63
ldr r5,=#(rcpapp-32)
ldrb r4,[r5,r4] @ u=lut5[x2-32]; // Q8
lsls r5,r3,#8
muls r5,r5,r4
asrs r5,#14 @ e=(i32)(u*(x<<8))>>14; // Q22
asrs r6,r5,#11
muls r6,r6,r6 @ e2=(e>>11)*(e>>11); // Q22
subs r5,r6
muls r5,r5,r4 @ c=(e-e2)*u; // Q30
lsls r6,r4,#7
asrs r5,#14
adds r5,#1
asrs r5,#1
subs r6,r5 @ u0=(u<<7)-((c+0x4000)>>15); // Q15
@ here
@ r0:r1 y mantissa
@ r2:r3 x mantissa
@ r6 u0, first approximation to 1/x Q15
@ r12: result sign, exponent
lsls r4,r3,#10
lsrs r5,r2,#22
orrs r5,r4 @ x1=(q30)x
muls r5,r6 @ u0*x1 Q45
asrs r5,#15 @ v=u0*x1-1 Q30
muls r5,r6 @ u0*v Q45
asrs r5,#14
adds r5,#1
asrs r5,#1 @ round u0*v to Q30
lsls r6,#15
subs r6,r5 @ u1 Q30
@ here
@ r0:r1 y mantissa
@ r2:r3 x mantissa
@ r6 u1, second approximation to 1/x Q30
@ r12: result sign, exponent
push {r2,r3}
lsls r4,r1,#11
lsrs r5,r0,#21
orrs r4,r5 @ y0=(q31)y
mul32_32_64 r4,r6, r4,r5, r2,r3,r7,r4,r5 @ y0*u1 Q61
adds r4,r4
adcs r5,r5 @ a0=(q30)(y0*u1)
@ here
@ r0:r1 y mantissa
@ r5 a0, first approximation to y/x Q30
@ r6 u1, second approximation to 1/x Q30
@ r12 result sign, exponent
ldr r2,[r13,#0] @ xL
mul32_32_64 r2,r5, r2,r3, r1,r4,r7,r2,r3 @ xL*a0
ldr r4,[r13,#4] @ xH
muls r4,r5 @ xH*a0
adds r3,r4 @ r2:r3 now x*a0 Q82
lsrs r2,#25
lsls r1,r3,#7
orrs r2,r1 @ r2 now x*a0 Q57; r7:r2 is x*a0 Q89
lsls r4,r0,#5 @ y Q57
subs r0,r4,r2 @ r0x=y-x*a0 Q57 (signed)
@ here
@ r0 r0x Q57
@ r5 a0, first approximation to y/x Q30
@ r4 yL Q57
@ r6 u1 Q30
@ r12 result sign, exponent
muls32_32_64 r0,r6, r7,r6, r1,r2,r3, r7,r6 @ r7:r6 r0x*u1 Q87
asrs r3,r6,#25
adds r5,r3
lsls r3,r6,#7 @ r3:r5 a1 Q62 (but bottom 7 bits are zero so 55 bits of precision after binary point)
@ here we could recover another 7 bits of precision (but not accuracy) from the top of r7
@ but these bits are thrown away in the rounding and conversion to Q52 below
@ here
@ r3:r5 a1 Q62 candidate quotient [0.5,2) or so
@ r4 yL Q57
@ r12 result sign, exponent
movs r6,#0
adds r3,#128 @ for initial rounding to Q53
adcs r5,r5,r6
lsrs r1,r5,#30
bne dd_0
@ here candidate quotient a1 is in range [0.5,1)
@ so 30 significant bits in r5
lsls r4,#1 @ y now Q58
lsrs r1,r5,#9 @ to Q52
lsls r0,r5,#23
lsrs r3,#9 @ 0.5ulp-significance bit in carry: if this is 1 we may need to correct result
orrs r0,r3
bcs dd_1
b dd_2
dd_0:
@ here candidate quotient a1 is in range [1,2)
@ so 31 significant bits in r5
movs r2,#4
add r12,r12,r2 @ fix exponent; r3:r5 now effectively Q61
adds r3,#128 @ complete rounding to Q53
adcs r5,r5,r6
lsrs r1,r5,#10
lsls r0,r5,#22
lsrs r3,#10 @ 0.5ulp-significance bit in carry: if this is 1 we may need to correct result
orrs r0,r3
bcc dd_2
dd_1:
@ here
@ r0:r1 rounded result Q53 [0.5,1) or Q52 [1,2), but may not be correctly rounded-to-nearest
@ r4 yL Q58 or Q57
@ r12 result sign, exponent
@ carry set
adcs r0,r0,r0
adcs r1,r1,r1 @ z Q53 with 1 in LSB
lsls r4,#16 @ Q105-32=Q73
ldr r2,[r13,#0] @ xL Q52
ldr r3,[r13,#4] @ xH Q20
movs r5,r1 @ zH Q21
muls r5,r2 @ zH*xL Q73
subs r4,r5
muls r3,r0 @ zL*xH Q73
subs r4,r3
mul32_32_64 r2,r0, r2,r3, r5,r6,r7,r2,r3 @ xL*zL
rsbs r2,#0 @ borrow from low half?
sbcs r4,r3 @ y-xz Q73 (remainder bits 52..73)
cmp r4,#0
bmi 1f
movs r2,#0 @ round up
adds r0,#1
adcs r1,r2
1:
lsrs r0,#1 @ shift back down to Q52
lsls r2,r1,#31
orrs r0,r2
lsrs r1,#1
dd_2:
add r13,#8
mov r2,r12
lsls r7,r2,#31 @ result sign
asrs r2,#2 @ result exponent
ldr r3,=#0x3fd
adds r2,r3
ldr r3,=#0x7fe
cmp r2,r3
bhs dd_3 @ over- or underflow?
lsls r2,#20
adds r1,r2 @ pack exponent
dd_5:
adds r1,r7 @ pack sign
pop {r4-r7,r15}
dd_3:
movs r0,#0
cmp r2,#0
bgt dd_4 @ overflow?
movs r1,r7
pop {r4-r7,r15}
dd_4:
adds r3,#1 @ 0x7ff
lsls r1,r3,#20
b dd_5
/*
Approach to square root x=sqrt(y) is as follows.
First generate a3, an approximation to 1/sqrt(y) to about 30 bits. Multiply this by y
to give a4~sqrt(y) to about 28 bits and a remainder r4=y-a4^2. Then, because
d sqrt(y) / dy = 1 / (2 sqrt(y)) let d4=r4*a3/2 and then the value a5=a4+d4 is
a better approximation to sqrt(y). If this is near a rounding boundary we
compute an exact remainder y-a5*a5 to decide whether to round up or down.
The calculation of a3 and a4 is as given in dsqrttest.c. That code verifies exhaustively
that | 1 - a3a4 | < 10*2^-32, | r4 | < 40*2^-32 and | r4/y | < 20*2^-32.
More precisely, with "y" representing y truncated to 30 binary places:
u=(q3)y; // 24-entry table
a0=(q8~)"1/sqrtq(x+x_ulp/2)"(u); // first approximation from table
p0=(q16)(a0*a0) * (q16)y;
r0=(q20)(p0-1);
dy0=(q15)(r0*a0); // Newton-Raphson correction term
a1=(q16)a0-dy0/2; // good to ~9 bits
p1=(q19)(a1*a1)*(q19)y;
r1=(q23)(p1-1);
dy1=(q15~)(r1*a1); // second Newton-Raphson correction
a2x=(q16)a1-dy1/2; // good to ~16 bits
a2=a2x-a2x/1t16; // prevent overflow of a2*a2 in 32 bits
p2=(a2*a2)*(q30)y; // Q62
r2=(q36)(p2-1+1t-31);
dy2=(q30)(r2*a2); // Q52->Q30
a3=(q31)a2-dy2/2; // good to about 30 bits
a4=(q30)(a3*(q30)y+1t-31); // good to about 28 bits
Error analysis
r₄ = y - a₄²
d₄ = 1/2 a₃r₄
a₅ = a₄ + d₄
r₅ = y - a₅²
= y - ( a₄ + d₄ )²
= y - a₄² - a₃a₄r₄ - 1/4 a₃²r₄²
= r₄ - a₃a₄r₄ - 1/4 a₃²r₄²
| r₅ | < | r₄ | | 1 - a₃a₄ | + 1/4 r₄²
a₅ = √y √( 1 - r₅/y )
= √y ( 1 - 1/2 r₅/y + ... )
So to first order (second order being very tiny)
√y - a₅ = 1/2 r₅/y
and
| √y - a₅ | < 1/2 ( | r₄/y | | 1 - a₃a₄ | + 1/4 r₄²/y )
From dsqrttest.c (conservatively):
< 1/2 ( 20*2^-32 * 10*2^-32 + 1/4 * 40*2^-32*20*2^-32 )
= 1/2 ( 200 + 200 ) * 2^-64
< 2^-56
Empirically we see about 1ulp worst-case error including rounding at Q57.
To determine correctly whether the exact remainder calculation can be skipped we need a result
accurate to < 0.25ulp at Q52, or 2^-54.
*/
dq_2:
bge dq_3 @ +Inf?
movs r1,#0
b dq_4
dq_0:
lsrs r1,#31
lsls r1,#31 @ preserve sign bit
lsrs r2,#21 @ extract exponent
beq dq_4 @ -0? return it
asrs r1,#11 @ make -Inf
b dq_4
dq_3:
ldr r1,=#0x7ff
lsls r1,#20 @ return +Inf
dq_4:
movs r0,#0
dq_1:
bx r14
.align 2
.thumb_func
qfp_dsqrt:
lsls r2,r1,#1
bcs dq_0 @ negative?
lsrs r2,#21 @ extract exponent
subs r2,#1
ldr r3,=#0x7fe
cmp r2,r3
bhs dq_2 @ catches 0 and +Inf
push {r4-r7,r14}
lsls r4,r2,#20
subs r1,r4 @ insert implied 1
lsrs r2,#1
bcc 1f @ even exponent? skip
adds r0,r0,r0 @ odd exponent: shift up mantissa
adcs r1,r1,r1
1:
lsrs r3,#2
adds r2,r3
lsls r2,#20
mov r12,r2 @ save result exponent
@ here
@ r0:r1 y mantissa Q52 [1,4)
@ r12 result exponent
adr r4,drsqrtapp-8 @ first eight table entries are never accessed because of the mantissa's leading 1
lsrs r2,r1,#17 @ y Q3
ldrb r2,[r4,r2] @ initial approximation to reciprocal square root a0 Q8
lsrs r3,r1,#4 @ first Newton-Raphson iteration
muls r3,r2
muls r3,r2 @ i32 p0=a0*a0*(y>>14); // Q32
asrs r3,r3,#12 @ i32 r0=p0>>12; // Q20
muls r3,r2
asrs r3,#13 @ i32 dy0=(r0*a0)>>13; // Q15
lsls r2,#8
subs r2,r3 @ i32 a1=(a0<<8)-dy0; // Q16
movs r3,r2
muls r3,r3
lsrs r3,#13
lsrs r4,r1,#1
muls r3,r4 @ i32 p1=((a1*a1)>>11)*(y>>11); // Q19*Q19=Q38
asrs r3,#15 @ i32 r1=p1>>15; // Q23
muls r3,r2
asrs r3,#23
adds r3,#1
asrs r3,#1 @ i32 dy1=(r1*a1+(1<<23))>>24; // Q23*Q16=Q39; Q15
subs r2,r3 @ i32 a2=a1-dy1; // Q16
lsrs r3,r2,#16
subs r2,r3 @ if(a2>=0x10000) a2=0xffff; to prevent overflow of a2*a2
@ here
@ r0:r1 y mantissa
@ r2 a2 ~ 1/sqrt(y) Q16
@ r12 result exponent
movs r3,r2
muls r3,r3
lsls r1,#10
lsrs r4,r0,#22
orrs r1,r4 @ y Q30
mul32_32_64 r1,r3, r4,r3, r5,r6,r7,r4,r3 @ i64 p2=(ui64)(a2*a2)*(ui64)y; // Q62 r4:r3
lsls r5,r3,#6
lsrs r4,#26
orrs r4,r5
adds r4,#0x20 @ i32 r2=(p2>>26)+0x20; // Q36 r4
uxth r5,r4
muls r5,r2
asrs r4,#16
muls r4,r2
lsrs r5,#16
adds r4,r5
asrs r4,#6 @ i32 dy2=((i64)r2*(i64)a2)>>22; // Q36*Q16=Q52; Q30
lsls r2,#15
subs r2,r4
@ here
@ r0 y low bits
@ r1 y Q30
@ r2 a3 ~ 1/sqrt(y) Q31
@ r12 result exponent
mul32_32_64 r2,r1, r3,r4, r5,r6,r7,r3,r4
adds r3,r3,r3
adcs r4,r4,r4
adds r3,r3,r3
movs r3,#0
adcs r3,r4 @ ui32 a4=((ui64)a3*(ui64)y+(1U<<31))>>31; // Q30
@ here
@ r0 y low bits
@ r1 y Q30
@ r2 a3 Q31 ~ 1/sqrt(y)
@ r3 a4 Q30 ~ sqrt(y)
@ r12 result exponent
square32_64 r3, r4,r5, r6,r5,r7
lsls r6,r0,#8
lsrs r7,r1,#2
subs r6,r4
sbcs r7,r5 @ r4=(q60)y-a4*a4
@ by exhaustive testing, r4 = fffffffc0e134fdc .. 00000003c2bf539c Q60
lsls r5,r7,#29
lsrs r6,#3
adcs r6,r5 @ r4 Q57 with rounding
muls32_32_64 r6,r2, r6,r2, r4,r5,r7,r6,r2 @ d4=a3*r4/2 Q89
@ r4+d4 is correct to 1ULP at Q57, tested on ~9bn cases including all extreme values of r4 for each possible y Q30
adds r2,#8
asrs r2,#5 @ d4 Q52, rounded to Q53 with spare bit in carry
@ here
@ r0 y low bits
@ r1 y Q30
@ r2 d4 Q52, rounded to Q53
@ C flag contains d4_b53
@ r3 a4 Q30
bcs dq_5
lsrs r5,r3,#10 @ a4 Q52
lsls r4,r3,#22
asrs r1,r2,#31
adds r0,r2,r4
adcs r1,r5 @ a4+d4
add r1,r12 @ pack exponent
pop {r4-r7,r15}
.ltorg
@ round(sqrt(2^22./[68:8:252]))
drsqrtapp:
.byte 0xf8,0xeb,0xdf,0xd6,0xcd,0xc5,0xbe,0xb8
.byte 0xb2,0xad,0xa8,0xa4,0xa0,0x9c,0x99,0x95
.byte 0x92,0x8f,0x8d,0x8a,0x88,0x85,0x83,0x81
dq_5:
@ here we are near a rounding boundary, C is set
adcs r2,r2,r2 @ d4 Q53+1ulp
lsrs r5,r3,#9
lsls r4,r3,#23 @ r4:r5 a4 Q53
asrs r1,r2,#31
adds r4,r2,r4
adcs r5,r1 @ r4:r5 a5=a4+d4 Q53+1ulp
movs r3,r5
muls r3,r4
square32_64 r4,r1,r2,r6,r2,r7
adds r2,r3
adds r2,r3 @ r1:r2 a5^2 Q106
lsls r0,#22 @ y Q84
rsbs r1,#0
sbcs r0,r2 @ remainder y-a5^2
bmi 1f @ y<a5^2: no need to increment a5
movs r3,#0
adds r4,#1
adcs r5,r3 @ bump a5 if over rounding boundary
1:
lsrs r0,r4,#1
lsrs r1,r5,#1
lsls r5,#31
orrs r0,r5
add r1,r12
pop {r4-r7,r15}
@ compare r0:r1 against r2:r3, returning -1/0/1 for <, =, >
@ also set flags accordingly
.thumb_func
qfp_dcmp:
push {r4,r6,r7,r14}
ldr r7,=#0x7ff @ flush NaNs and denormals
lsls r4,r1,#1
lsrs r4,#21
beq 1f
cmp r4,r7
bne 2f
1:
movs r0,#0
lsrs r1,#20
lsls r1,#20
2:
lsls r4,r3,#1
lsrs r4,#21
beq 1f
cmp r4,r7
bne 2f
1:
movs r2,#0
lsrs r3,#20
lsls r3,#20
2:
dcmp_fast_entry:
movs r6,#1
eors r3,r1
bmi 4f @ opposite signs? then can proceed on basis of sign of x
eors r3,r1 @ restore r3
bpl 1f
rsbs r6,#0 @ negative? flip comparison
1:
cmp r1,r3
bne 1f
cmp r0,r2
bhi 2f
blo 3f
5:
movs r6,#0 @ equal? result is 0
1:
bgt 2f
3:
rsbs r6,#0
2:
subs r0,r6,#0 @ copy and set flags
pop {r4,r6,r7,r15}
4:
orrs r3,r1 @ make -0==+0
adds r3,r3
orrs r3,r0
orrs r3,r2
beq 5b
cmp r1,#0
bge 2b
b 3b
@ "scientific" functions start here
.thumb_func
push_r8_r11:
mov r4,r8
mov r5,r9
mov r6,r10
mov r7,r11
push {r4-r7}
bx r14
.thumb_func
pop_r8_r11:
pop {r4-r7}
mov r8,r4
mov r9,r5
mov r10,r6
mov r11,r7
bx r14
@ double-length CORDIC rotation step
@ r0:r1 ω
@ r6 32-i (complementary shift)
@ r7 i (shift)
@ r8:r9 x
@ r10:r11 y
@ r12 coefficient pointer
@ an option in rotation mode would be to compute the sequence of σ values
@ in one pass, rotate the initial vector by the residual ω and then run a
@ second pass to compute the final x and y. This would relieve pressure
@ on registers and hence possibly be faster. The same trick does not work
@ in vectoring mode (but perhaps one could work to single precision in
@ a first pass and then double precision in a second pass?).
.thumb_func
dcordic_vec_step:
mov r2,r12
ldmia r2!,{r3,r4}
mov r12,r2
mov r2,r11
cmp r2,#0
blt 1f
b 2f
.thumb_func
dcordic_rot_step:
mov r2,r12
ldmia r2!,{r3,r4}
mov r12,r2
cmp r1,#0
bge 1f
2:
@ ω<0 / y>=0
@ ω+=dω
@ x+=y>>i, y-=x>>i
adds r0,r3
adcs r1,r4
mov r3,r11
asrs r3,r7
mov r4,r11
lsls r4,r6
mov r2,r10
lsrs r2,r7
orrs r2,r4 @ r2:r3 y>>i, rounding in carry
mov r4,r8
mov r5,r9 @ r4:r5 x
adcs r2,r4
adcs r3,r5 @ r2:r3 x+(y>>i)
mov r8,r2
mov r9,r3
mov r3,r5
lsls r3,r6
asrs r5,r7
lsrs r4,r7
orrs r4,r3 @ r4:r5 x>>i, rounding in carry
mov r2,r10
mov r3,r11
sbcs r2,r4
sbcs r3,r5 @ r2:r3 y-(x>>i)
mov r10,r2
mov r11,r3
bx r14
@ ω>0 / y<0
@ ω-=dω
@ x-=y>>i, y+=x>>i
1:
subs r0,r3
sbcs r1,r4
mov r3,r9
asrs r3,r7
mov r4,r9
lsls r4,r6
mov r2,r8
lsrs r2,r7
orrs r2,r4 @ r2:r3 x>>i, rounding in carry
mov r4,r10
mov r5,r11 @ r4:r5 y
adcs r2,r4
adcs r3,r5 @ r2:r3 y+(x>>i)
mov r10,r2
mov r11,r3
mov r3,r5
lsls r3,r6
asrs r5,r7
lsrs r4,r7
orrs r4,r3 @ r4:r5 y>>i, rounding in carry
mov r2,r8
mov r3,r9
sbcs r2,r4
sbcs r3,r5 @ r2:r3 x-(y>>i)
mov r8,r2
mov r9,r3
bx r14
ret_dzero:
movs r0,#0
movs r1,#0
bx r14
@ convert double to signed int, rounding towards 0, clamping
.thumb_func
qfp_double2int_z:
cmp r1,#0
bge qfp_double2int @ +ve or zero? then use rounding towards -Inf
push {r14}
lsls r1,#1 @ -ve: clear sign bit
lsrs r1,#1
bl qfp_double2uint @ convert to unsigned, rounding towards -Inf
movs r1,#1
lsls r1,#31 @ r1=0x80000000
cmp r0,r1
bhi 1f
rsbs r0,#0
pop {r15}
1:
mov r0,r1
pop {r15}
@ convert packed double in r0:r1 to signed/unsigned 32/64-bit integer/fixed-point value in r0:r1 [with r2 places after point], with rounding towards -Inf
@ fixed-point versions only work with reasonable values in r2 because of the way dunpacks works
.thumb_func
qfp_double2int:
movs r2,#0 @ and fall through
.thumb_func
qfp_double2fix:
push {r14}
adds r2,#32
bl qfp_double2fix64
movs r0,r1
pop {r15}
.thumb_func
qfp_double2uint:
movs r2,#0 @ and fall through
.thumb_func
qfp_double2ufix:
push {r14}
adds r2,#32
bl qfp_double2ufix64
movs r0,r1
pop {r15}
.thumb_func
qfp_float2int64_z:
cmp r0,#0
bge qfp_float2int64 @ +ve or zero? then use rounding towards -Inf
push {r14}
lsls r0,#1 @ -ve: clear sign bit
lsrs r0,#1
bl qfp_float2uint64 @ convert to unsigned, rounding towards -Inf
movs r2,#1
lsls r2,#31 @ r2=0x80000000
cmp r1,r2
bhs 1f
mvns r1,r1
rsbs r0,#0
bcc 2f
adds r1,#1
2:
pop {r15}
1:
movs r0,#0
mov r1,r2
pop {r15}
.thumb_func
qfp_float2int64:
movs r1,#0 @ and fall through
.thumb_func
qfp_float2fix64:
push {r14}
bl f2fix
b d2f64_a
.thumb_func
qfp_float2uint64:
movs r1,#0 @ and fall through
.thumb_func
qfp_float2ufix64:
asrs r3,r0,#23 @ negative? return 0
bmi ret_dzero
@ and fall through
@ convert float in r0 to signed fixed point in r0:r1:r3, r1 places after point, rounding towards -Inf
@ result clamped so that r3 can only be 0 or -1
@ trashes r12
.thumb_func
f2fix:
push {r4,r14}
mov r12,r1
asrs r3,r0,#31
lsls r0,#1
lsrs r2,r0,#24
beq 1f @ zero?
cmp r2,#0xff @ Inf?
beq 2f
subs r1,r2,#1
subs r2,#0x7f @ remove exponent bias
lsls r1,#24
subs r0,r1 @ insert implied 1
eors r0,r3
subs r0,r3 @ top two's complement
asrs r1,r0,#4 @ convert to double format
lsls r0,#28
b d2fix_a
1:
movs r0,#0
movs r1,r0
movs r3,r0
pop {r4,r15}
2:
mvns r0,r3 @ return max/min value
mvns r1,r3
pop {r4,r15}
.thumb_func
qfp_double2int64_z:
cmp r1,#0
bge qfp_double2int64 @ +ve or zero? then use rounding towards -Inf
push {r14}
lsls r1,#1 @ -ve: clear sign bit
lsrs r1,#1
bl qfp_double2uint64 @ convert to unsigned, rounding towards -Inf
cmp r1,#0
blt 1f
mvns r1,r1
rsbs r0,#0
bcc 2f
adds r1,#1
2:
pop {r15}
1:
movs r0,#0
movs r1,#1
lsls r1,#31 @ 0x80000000
pop {r15}
.thumb_func
qfp_double2int64:
movs r2,#0 @ and fall through
.thumb_func
qfp_double2fix64:
push {r14}
bl d2fix
d2f64_a:
asrs r2,r1,#31
cmp r2,r3
bne 1f @ sign extension bits fail to match sign of result?
pop {r15}
1:
mvns r0,r3
movs r1,#1
lsls r1,#31
eors r1,r1,r0 @ generate extreme fixed-point values
pop {r15}
.thumb_func
qfp_double2uint64:
movs r2,#0 @ and fall through
.thumb_func
qfp_double2ufix64:
asrs r3,r1,#20 @ negative? return 0
bmi ret_dzero
@ and fall through
@ convert double in r0:r1 to signed fixed point in r0:r1:r3, r2 places after point, rounding towards -Inf
@ result clamped so that r3 can only be 0 or -1
@ trashes r12
.thumb_func
d2fix:
push {r4,r14}
mov r12,r2
bl dunpacks
asrs r4,r2,#16
adds r4,#1
bge 1f
movs r1,#0 @ -0 -> +0
1:
asrs r3,r1,#31
d2fix_a:
@ here
@ r0:r1 two's complement mantissa
@ r2 unbaised exponent
@ r3 mantissa sign extension bits
add r2,r12 @ exponent plus offset for required binary point position
subs r2,#52 @ required shift
bmi 1f @ shift down?
@ here a shift up by r2 places
cmp r2,#12 @ will clamp?
bge 2f
movs r4,r0
lsls r1,r2
lsls r0,r2
rsbs r2,#0
adds r2,#32 @ complementary shift
lsrs r4,r2
orrs r1,r4
pop {r4,r15}
2:
mvns r0,r3
mvns r1,r3 @ overflow: clamp to extreme fixed-point values
pop {r4,r15}
1:
@ here a shift down by -r2 places
adds r2,#32
bmi 1f @ long shift?
mov r4,r1
lsls r4,r2
rsbs r2,#0
adds r2,#32 @ complementary shift
asrs r1,r2
lsrs r0,r2
orrs r0,r4
pop {r4,r15}
1:
@ here a long shift down
movs r0,r1
asrs r1,#31 @ shift down 32 places
adds r2,#32
bmi 1f @ very long shift?
rsbs r2,#0
adds r2,#32
asrs r0,r2
pop {r4,r15}
1:
movs r0,r3 @ result very near zero: use sign extension bits
movs r1,r3
pop {r4,r15}
@ float <-> double conversions
.thumb_func
qfp_float2double:
lsrs r3,r0,#31 @ sign bit
lsls r3,#31
lsls r1,r0,#1
lsrs r2,r1,#24 @ exponent
beq 1f @ zero?
cmp r2,#0xff @ Inf?
beq 2f
lsrs r1,#4 @ exponent and top 20 bits of mantissa
ldr r2,=#(0x3ff-0x7f)<<20 @ difference in exponent offsets
adds r1,r2
orrs r1,r3
lsls r0,#29 @ bottom 3 bits of mantissa
bx r14
1:
movs r1,r3 @ return signed zero
3:
movs r0,#0
bx r14
2:
ldr r1,=#0x7ff00000 @ return signed infinity
adds r1,r3
b 3b
.thumb_func
qfp_double2float:
lsls r2,r1,#1
lsrs r2,#21 @ exponent
ldr r3,=#0x3ff-0x7f
subs r2,r3 @ fix exponent bias
ble 1f @ underflow or zero
cmp r2,#0xff
bge 2f @ overflow or infinity
lsls r2,#23 @ position exponent of result
lsrs r3,r1,#31
lsls r3,#31
orrs r2,r3 @ insert sign
lsls r3,r0,#3 @ rounding bits
lsrs r0,#29
lsls r1,#12
lsrs r1,#9
orrs r0,r1 @ assemble mantissa
orrs r0,r2 @ insert exponent and sign
lsls r3,#1
bcc 3f @ no rounding
beq 4f @ all sticky bits 0?
5:
adds r0,#1
3:
bx r14
4:
lsrs r3,r0,#1 @ odd? then round up
bcs 5b
bx r14
1:
beq 6f @ check case where value is just less than smallest normal
7:
lsrs r0,r1,#31
lsls r0,#31
bx r14
6:
lsls r2,r1,#12 @ 20 1:s at top of mantissa?
asrs r2,#12
adds r2,#1
bne 7b
lsrs r2,r0,#29 @ and 3 more 1:s?
cmp r2,#7
bne 7b
movs r2,#1 @ return smallest normal with correct sign
b 8f
2:
movs r2,#0xff
8:
lsrs r0,r1,#31 @ return signed infinity
lsls r0,#8
adds r0,r2
lsls r0,#23
bx r14
@ convert signed/unsigned 32/64-bit integer/fixed-point value in r0:r1 [with r2 places after point] to packed double in r0:r1, with rounding
.thumb_func
qfp_uint2double:
movs r1,#0 @ and fall through
.thumb_func
qfp_ufix2double:
movs r2,r1
movs r1,#0
b qfp_ufix642double
.thumb_func
qfp_int2double:
movs r1,#0 @ and fall through
.thumb_func
qfp_fix2double:
movs r2,r1
asrs r1,r0,#31 @ sign extend
b qfp_fix642double
.thumb_func
qfp_uint642double:
movs r2,#0 @ and fall through
.thumb_func
qfp_ufix642double:
movs r3,#0
b uf2d
.thumb_func
qfp_int642double:
movs r2,#0 @ and fall through
.thumb_func
qfp_fix642double:
asrs r3,r1,#31 @ sign bit across all bits
eors r0,r3
eors r1,r3
subs r0,r3
sbcs r1,r3
uf2d:
push {r4,r5,r14}
ldr r4,=#0x432
subs r2,r4,r2 @ form biased exponent
@ here
@ r0:r1 unnormalised mantissa
@ r2 -Q (will become exponent)
@ r3 sign across all bits
cmp r1,#0
bne 1f @ short normalising shift?
movs r1,r0
beq 2f @ zero? return it
movs r0,#0
subs r2,#32 @ fix exponent
1:
asrs r4,r1,#21
bne 3f @ will need shift down (and rounding?)
bcs 4f @ normalised already?
5:
subs r2,#1
adds r0,r0 @ shift up
adcs r1,r1
lsrs r4,r1,#21
bcc 5b
4:
ldr r4,=#0x7fe
cmp r2,r4
bhs 6f @ over/underflow? return signed zero/infinity
7:
lsls r2,#20 @ pack and return
adds r1,r2
lsls r3,#31
adds r1,r3
2:
pop {r4,r5,r15}
6: @ return signed zero/infinity according to unclamped exponent in r2
mvns r2,r2
lsrs r2,#21
movs r0,#0
movs r1,#0
b 7b
3:
@ here we need to shift down to normalise and possibly round
bmi 1f @ already normalised to Q63?
2:
subs r2,#1
adds r0,r0 @ shift up
adcs r1,r1
bpl 2b
1:
@ here we have a 1 in b63 of r0:r1
adds r2,#11 @ correct exponent for subsequent shift down
lsls r4,r0,#21 @ save bits for rounding
lsrs r0,#11
lsls r5,r1,#21
orrs r0,r5
lsrs r1,#11
lsls r4,#1
beq 1f @ sticky bits are zero?
8:
movs r4,#0
adcs r0,r4
adcs r1,r4
b 4b
1:
bcc 4b @ sticky bits are zero but not on rounding boundary
lsrs r4,r0,#1 @ increment if odd (force round to even)
b 8b
.ltorg
.thumb_func
dunpacks:
mdunpacks r0,r1,r2,r3,r4
ldr r3,=#0x3ff
subs r2,r3 @ exponent without offset
bx r14
@ r0:r1 signed mantissa Q52
@ r2 unbiased exponent < 10 (i.e., |x|<2^10)
@ r4 pointer to:
@ - divisor reciprocal approximation r=1/d Q15
@ - divisor d Q62 0..20
@ - divisor d Q62 21..41
@ - divisor d Q62 42..62
@ returns:
@ r0:r1 reduced result y Q62, -0.6 d < y < 0.6 d (better in practice)
@ r2 quotient q (number of reductions)
@ if exponent >=10, returns r0:r1=0, r2=1024*mantissa sign
@ designed to work for 0.5<d<2, in particular d=ln2 (~0.7) and d=π/2 (~1.6)
.thumb_func
dreduce:
adds r2,#2 @ e+2
bmi 1f @ |x|<0.25, too small to need adjustment
cmp r2,#12
bge 4f
2:
movs r5,#17
subs r5,r2 @ 15-e
movs r3,r1 @ Q20
asrs r3,r5 @ x Q5
adds r2,#8 @ e+10
adds r5,#7 @ 22-e = 32-(e+10)
movs r6,r0
lsrs r6,r5
lsls r0,r2
lsls r1,r2
orrs r1,r6 @ r0:r1 x Q62
ldmia r4,{r4-r7}
muls r3,r4 @ rx Q20
asrs r2,r3,#20
movs r3,#0
adcs r2,r3 @ rx Q0 rounded = q; for e.g. r=1.5 |q|<1.5*2^10
muls r5,r2 @ qd in pieces: L Q62
muls r6,r2 @ M Q41
muls r7,r2 @ H Q20
lsls r7,#10
asrs r4,r6,#11
lsls r6,#21
adds r6,r5
adcs r7,r4
asrs r5,#31
adds r7,r5 @ r6:r7 qd Q62
subs r0,r6
sbcs r1,r7 @ remainder Q62
bx r14
4:
movs r2,#12 @ overflow: clamp to +/-1024
movs r0,#0
asrs r1,#31
lsls r1,#1
adds r1,#1
lsls r1,#20
b 2b
1:
lsls r1,#8
lsrs r3,r0,#24
orrs r1,r3
lsls r0,#8 @ r0:r1 Q60, to be shifted down -r2 places
rsbs r3,r2,#0
adds r2,#32 @ shift down in r3, complementary shift in r2
bmi 1f @ long shift?
2:
movs r4,r1
asrs r1,r3
lsls r4,r2
lsrs r0,r3
orrs r0,r4
movs r2,#0 @ rounding
adcs r0,r2
adcs r1,r2
bx r14
1:
movs r0,r1 @ down 32 places
asrs r1,#31
subs r3,#32
adds r2,#32
bpl 2b
movs r0,#0 @ very long shift? return 0
movs r1,#0
movs r2,#0
bx r14
.thumb_func
qfp_dtan:
push {r4-r7,r14}
bl push_r8_r11
bl dsincos
mov r12,r0 @ save ε
bl dcos_finish
push {r0,r1}
mov r0,r12
bl dsin_finish
pop {r2,r3}
bl pop_r8_r11
b ddiv0 @ compute sin θ/cos θ
.thumb_func
qfp_dcos:
push {r4-r7,r14}
bl push_r8_r11
bl dsincos
bl dcos_finish
b 1f
.thumb_func
qfp_dsin:
push {r4-r7,r14}
bl push_r8_r11
bl dsincos
bl dsin_finish
1:
bl pop_r8_r11
pop {r4-r7,r15}
@ unpack double θ in r0:r1, range reduce and calculate ε, cos α and sin α such that
@ θ=α+ε and |ε|≤2^-32
@ on return:
@ r0:r1 ε (residual ω, where θ=α+ε) Q62, |ε|≤2^-32 (so fits in r0)
@ r8:r9 cos α Q62
@ r10:r11 sin α Q62
.thumb_func
dsincos:
push {r14}
bl dunpacks
adr r4,dreddata0
bl dreduce
movs r4,#0
ldr r5,=#0x9df04dbb @ this value compensates for the non-unity scaling of the CORDIC rotations
ldr r6,=#0x36f656c5
lsls r2,#31
bcc 1f
@ quadrant 2 or 3
mvns r6,r6
rsbs r5,r5,#0
adcs r6,r4
1:
lsls r2,#1
bcs 1f
@ even quadrant
mov r10,r4
mov r11,r4
mov r8,r5
mov r9,r6
b 2f
1:
@ odd quadrant
mov r8,r4
mov r9,r4
mov r10,r5
mov r11,r6
2:
adr r4,dtab_cc
mov r12,r4
movs r7,#1
movs r6,#31
1:
bl dcordic_rot_step
adds r7,#1
subs r6,#1
cmp r7,#33
bne 1b
pop {r15}
dcos_finish:
@ here
@ r0:r1 ε (residual ω, where θ=α+ε) Q62, |ε|≤2^-32 (so fits in r0)
@ r8:r9 cos α Q62
@ r10:r11 sin α Q62
@ and we wish to calculate cos θ=cos(α+ε)~cos α - ε sin α
mov r1,r11
@ mov r2,r10
@ lsrs r2,#31
@ adds r1,r2 @ rounding improves accuracy very slightly
muls32_s32_64 r0,r1, r2,r3, r4,r5,r6,r2,r3
@ r2:r3 ε sin α Q(62+62-32)=Q92
mov r0,r8
mov r1,r9
lsls r5,r3,#2
asrs r3,r3,#30
lsrs r2,r2,#30
orrs r2,r5
sbcs r0,r2 @ include rounding
sbcs r1,r3
movs r2,#62
b qfp_fix642double
dsin_finish:
@ here
@ r0:r1 ε (residual ω, where θ=α+ε) Q62, |ε|≤2^-32 (so fits in r0)
@ r8:r9 cos α Q62
@ r10:r11 sin α Q62
@ and we wish to calculate sin θ=sin(α+ε)~sin α + ε cos α
mov r1,r9
muls32_s32_64 r0,r1, r2,r3, r4,r5,r6,r2,r3
@ r2:r3 ε cos α Q(62+62-32)=Q92
mov r0,r10
mov r1,r11
lsls r5,r3,#2
asrs r3,r3,#30
lsrs r2,r2,#30
orrs r2,r5
adcs r0,r2 @ include rounding
adcs r1,r3
movs r2,#62
b qfp_fix642double
.ltorg
.align 2
dreddata0:
.word 0x0000517d @ 2/π Q15
.word 0x0014611A @ π/2 Q62=6487ED5110B4611A split into 21-bit pieces
.word 0x000A8885
.word 0x001921FB
.thumb_func
qfp_datan2:
@ r0:r1 y
@ r2:r3 x
push {r4-r7,r14}
bl push_r8_r11
ldr r5,=#0x7ff00000
movs r4,r1
ands r4,r5 @ y==0?
beq 1f
cmp r4,r5 @ or Inf/NaN?
bne 2f
1:
lsrs r1,#20 @ flush
lsls r1,#20
movs r0,#0
2:
movs r4,r3
ands r4,r5 @ x==0?
beq 1f
cmp r4,r5 @ or Inf/NaN?
bne 2f
1:
lsrs r3,#20 @ flush
lsls r3,#20
movs r2,#0
2:
movs r6,#0 @ quadrant offset
lsls r5,#11 @ constant 0x80000000
cmp r3,#0
bpl 1f @ skip if x positive
movs r6,#2
eors r3,r5
eors r1,r5
bmi 1f @ quadrant offset=+2 if y was positive
rsbs r6,#0 @ quadrant offset=-2 if y was negative
1:
@ now in quadrant 0 or 3
adds r7,r1,r5 @ r7=-r1
bpl 1f
@ y>=0: in quadrant 0
cmp r1,r3
ble 2f @ y<~x so 0≤θ<~π/4: skip
adds r6,#1
eors r1,r5 @ negate x
b 3f @ and exchange x and y = rotate by -π/2
1:
cmp r3,r7
bge 2f @ -y<~x so -π/4<~θ≤0: skip
subs r6,#1
eors r3,r5 @ negate y and ...
3:
movs r7,r0 @ exchange x and y
movs r0,r2
movs r2,r7
movs r7,r1
movs r1,r3
movs r3,r7
2:
@ here -π/4<~θ<~π/4
@ r6 has quadrant offset
push {r6}
cmp r2,#0
bne 1f
cmp r3,#0
beq 10f @ x==0 going into division?
lsls r4,r3,#1
asrs r4,#21
adds r4,#1
bne 1f @ x==Inf going into division?
lsls r4,r1,#1
asrs r4,#21
adds r4,#1 @ y also ±Inf?
bne 10f
subs r1,#1 @ make them both just finite
subs r3,#1
b 1f
10:
movs r0,#0
movs r1,#0
b 12f
1:
bl qfp_ddiv
movs r2,#62
bl qfp_double2fix64
@ r0:r1 y/x
mov r10,r0
mov r11,r1
movs r0,#0 @ ω=0
movs r1,#0
mov r8,r0
movs r2,#1
lsls r2,#30
mov r9,r2 @ x=1
adr r4,dtab_cc
mov r12,r4
movs r7,#1
movs r6,#31
1:
bl dcordic_vec_step
adds r7,#1
subs r6,#1
cmp r7,#33
bne 1b
@ r0:r1 atan(y/x) Q62
@ r8:r9 x residual Q62
@ r10:r11 y residual Q62
mov r2,r9
mov r3,r10
subs r2,#12 @ this makes atan(0)==0
@ the following is basically a division residual y/x ~ atan(residual y/x)
movs r4,#1
lsls r4,#29
movs r7,#0
2:
lsrs r2,#1
movs r3,r3 @ preserve carry
bmi 1f
sbcs r3,r2
adds r0,r4
adcs r1,r7
lsrs r4,#1
bne 2b
b 3f
1:
adcs r3,r2
subs r0,r4
sbcs r1,r7
lsrs r4,#1
bne 2b
3:
lsls r6,r1,#31
asrs r1,#1
lsrs r0,#1
orrs r0,r6 @ Q61
12:
pop {r6}
cmp r6,#0
beq 1f
ldr r4,=#0x885A308D @ π/2 Q61
ldr r5,=#0x3243F6A8
bpl 2f
mvns r4,r4 @ negative quadrant offset
mvns r5,r5
2:
lsls r6,#31
bne 2f @ skip if quadrant offset is ±1
adds r0,r4
adcs r1,r5
2:
adds r0,r4
adcs r1,r5
1:
movs r2,#61
bl qfp_fix642double
bl pop_r8_r11
pop {r4-r7,r15}
.ltorg
dtab_cc:
.word 0x61bb4f69, 0x1dac6705 @ atan 2^-1 Q62
.word 0x96406eb1, 0x0fadbafc @ atan 2^-2 Q62
.word 0xab0bdb72, 0x07f56ea6 @ atan 2^-3 Q62
.word 0xe59fbd39, 0x03feab76 @ atan 2^-4 Q62
.word 0xba97624b, 0x01ffd55b @ atan 2^-5 Q62
.word 0xdddb94d6, 0x00fffaaa @ atan 2^-6 Q62
.word 0x56eeea5d, 0x007fff55 @ atan 2^-7 Q62
.word 0xaab7776e, 0x003fffea @ atan 2^-8 Q62
.word 0x5555bbbc, 0x001ffffd @ atan 2^-9 Q62
.word 0xaaaaadde, 0x000fffff @ atan 2^-10 Q62
.word 0xf555556f, 0x0007ffff @ atan 2^-11 Q62
.word 0xfeaaaaab, 0x0003ffff @ atan 2^-12 Q62
.word 0xffd55555, 0x0001ffff @ atan 2^-13 Q62
.word 0xfffaaaab, 0x0000ffff @ atan 2^-14 Q62
.word 0xffff5555, 0x00007fff @ atan 2^-15 Q62
.word 0xffffeaab, 0x00003fff @ atan 2^-16 Q62
.word 0xfffffd55, 0x00001fff @ atan 2^-17 Q62
.word 0xffffffab, 0x00000fff @ atan 2^-18 Q62
.word 0xfffffff5, 0x000007ff @ atan 2^-19 Q62
.word 0xffffffff, 0x000003ff @ atan 2^-20 Q62
.word 0x00000000, 0x00000200 @ atan 2^-21 Q62 @ consider optimising these
.word 0x00000000, 0x00000100 @ atan 2^-22 Q62
.word 0x00000000, 0x00000080 @ atan 2^-23 Q62
.word 0x00000000, 0x00000040 @ atan 2^-24 Q62
.word 0x00000000, 0x00000020 @ atan 2^-25 Q62
.word 0x00000000, 0x00000010 @ atan 2^-26 Q62
.word 0x00000000, 0x00000008 @ atan 2^-27 Q62
.word 0x00000000, 0x00000004 @ atan 2^-28 Q62
.word 0x00000000, 0x00000002 @ atan 2^-29 Q62
.word 0x00000000, 0x00000001 @ atan 2^-30 Q62
.word 0x80000000, 0x00000000 @ atan 2^-31 Q62
.word 0x40000000, 0x00000000 @ atan 2^-32 Q62
.thumb_func
qfp_dexp:
push {r4-r7,r14}
bl dunpacks
adr r4,dreddata1
bl dreduce
cmp r1,#0
bge 1f
ldr r4,=#0xF473DE6B
ldr r5,=#0x2C5C85FD @ ln2 Q62
adds r0,r4
adcs r1,r5
subs r2,#1
1:
push {r2}
movs r7,#1 @ shift
adr r6,dtab_exp
movs r2,#0
movs r3,#1
lsls r3,#30 @ x=1 Q62
3:
ldmia r6!,{r4,r5}
mov r12,r6
subs r0,r4
sbcs r1,r5
bmi 1f
rsbs r6,r7,#0
adds r6,#32 @ complementary shift
movs r5,r3
asrs r5,r7
movs r4,r3
lsls r4,r6
movs r6,r2
lsrs r6,r7 @ rounding bit in carry
orrs r4,r6
adcs r2,r4
adcs r3,r5 @ x+=x>>i
b 2f
1:
adds r0,r4 @ restore argument
adcs r1,r5
2:
mov r6,r12
adds r7,#1
cmp r7,#33
bne 3b
@ here
@ r0:r1 ε (residual x, where x=a+ε) Q62, |ε|≤2^-32 (so fits in r0)
@ r2:r3 exp a Q62
@ and we wish to calculate exp x=exp a exp ε~(exp a)(1+ε)
muls32_32_64 r0,r3, r4,r1, r5,r6,r7,r4,r1
@ r4:r1 ε exp a Q(62+62-32)=Q92
lsrs r4,#30
lsls r0,r1,#2
orrs r0,r4
asrs r1,#30
adds r0,r2
adcs r1,r3
pop {r2}
rsbs r2,#0
adds r2,#62
bl qfp_fix642double @ in principle we can pack faster than this because we know the exponent
pop {r4-r7,r15}
.ltorg
.thumb_func
qfp_dln:
push {r4-r7,r14}
lsls r7,r1,#1
bcs 5f @ <0 ...
asrs r7,#21
beq 5f @ ... or =0? return -Inf
adds r7,#1
beq 6f @ Inf/NaN? return +Inf
bl dunpacks
push {r2}
lsls r1,#9
lsrs r2,r0,#23
orrs r1,r2
lsls r0,#9
@ r0:r1 m Q61 = m/2 Q62 0.5≤m/2<1
movs r7,#1 @ shift
adr r6,dtab_exp
mov r12,r6
movs r2,#0
movs r3,#0 @ y=0 Q62
3:
rsbs r6,r7,#0
adds r6,#32 @ complementary shift
movs r5,r1
asrs r5,r7
movs r4,r1
lsls r4,r6
movs r6,r0
lsrs r6,r7
orrs r4,r6 @ x>>i, rounding bit in carry
adcs r4,r0
adcs r5,r1 @ x+(x>>i)
lsrs r6,r5,#30
bne 1f @ x+(x>>i)>1?
movs r0,r4
movs r1,r5 @ x+=x>>i
mov r6,r12
ldmia r6!,{r4,r5}
subs r2,r4
sbcs r3,r5
1:
movs r4,#8
add r12,r4
adds r7,#1
cmp r7,#33
bne 3b
@ here:
@ r0:r1 residual x, nearly 1 Q62
@ r2:r3 y ~ ln m/2 = ln m - ln2 Q62
@ result is y + ln2 + ln x ~ y + ln2 + (x-1)
lsls r1,#2
asrs r1,#2 @ x-1
adds r2,r0
adcs r3,r1
pop {r7}
@ here:
@ r2:r3 ln m/2 = ln m - ln2 Q62
@ r7 unbiased exponent
adr r4,dreddata1+4
ldmia r4,{r0,r1,r4}
adds r7,#1
muls r0,r7 @ Q62
muls r1,r7 @ Q41
muls r4,r7 @ Q20
lsls r7,r1,#21
asrs r1,#11
asrs r5,r1,#31
adds r0,r7
adcs r1,r5
lsls r7,r4,#10
asrs r4,#22
asrs r5,r1,#31
adds r1,r7
adcs r4,r5
@ r0:r1:r4 exponent*ln2 Q62
asrs r5,r3,#31
adds r0,r2
adcs r1,r3
adcs r4,r5
@ r0:r1:r4 result Q62
movs r2,#62
1:
asrs r5,r1,#31
cmp r4,r5
beq 2f @ r4 a sign extension of r1?
lsrs r0,#4 @ no: shift down 4 places and try again
lsls r6,r1,#28
orrs r0,r6
lsrs r1,#4
lsls r6,r4,#28
orrs r1,r6
asrs r4,#4
subs r2,#4
b 1b
2:
bl qfp_fix642double
pop {r4-r7,r15}
5:
ldr r1,=#0xfff00000
movs r0,#0
pop {r4-r7,r15}
6:
ldr r1,=#0x7ff00000
movs r0,#0
pop {r4-r7,r15}
.ltorg
.align 2
dreddata1:
.word 0x0000B8AA @ 1/ln2 Q15
.word 0x0013DE6B @ ln2 Q62 Q62=2C5C85FDF473DE6B split into 21-bit pieces
.word 0x000FEFA3
.word 0x000B1721
dtab_exp:
.word 0xbf984bf3, 0x19f323ec @ log 1+2^-1 Q62
.word 0xcd4d10d6, 0x0e47fbe3 @ log 1+2^-2 Q62
.word 0x8abcb97a, 0x0789c1db @ log 1+2^-3 Q62
.word 0x022c54cc, 0x03e14618 @ log 1+2^-4 Q62
.word 0xe7833005, 0x01f829b0 @ log 1+2^-5 Q62
.word 0x87e01f1e, 0x00fe0545 @ log 1+2^-6 Q62
.word 0xac419e24, 0x007f80a9 @ log 1+2^-7 Q62
.word 0x45621781, 0x003fe015 @ log 1+2^-8 Q62
.word 0xa9ab10e6, 0x001ff802 @ log 1+2^-9 Q62
.word 0x55455888, 0x000ffe00 @ log 1+2^-10 Q62
.word 0x0aa9aac4, 0x0007ff80 @ log 1+2^-11 Q62
.word 0x01554556, 0x0003ffe0 @ log 1+2^-12 Q62
.word 0x002aa9ab, 0x0001fff8 @ log 1+2^-13 Q62
.word 0x00055545, 0x0000fffe @ log 1+2^-14 Q62
.word 0x8000aaaa, 0x00007fff @ log 1+2^-15 Q62
.word 0xe0001555, 0x00003fff @ log 1+2^-16 Q62
.word 0xf80002ab, 0x00001fff @ log 1+2^-17 Q62
.word 0xfe000055, 0x00000fff @ log 1+2^-18 Q62
.word 0xff80000b, 0x000007ff @ log 1+2^-19 Q62
.word 0xffe00001, 0x000003ff @ log 1+2^-20 Q62
.word 0xfff80000, 0x000001ff @ log 1+2^-21 Q62
.word 0xfffe0000, 0x000000ff @ log 1+2^-22 Q62
.word 0xffff8000, 0x0000007f @ log 1+2^-23 Q62
.word 0xffffe000, 0x0000003f @ log 1+2^-24 Q62
.word 0xfffff800, 0x0000001f @ log 1+2^-25 Q62
.word 0xfffffe00, 0x0000000f @ log 1+2^-26 Q62
.word 0xffffff80, 0x00000007 @ log 1+2^-27 Q62
.word 0xffffffe0, 0x00000003 @ log 1+2^-28 Q62
.word 0xfffffff8, 0x00000001 @ log 1+2^-29 Q62
.word 0xfffffffe, 0x00000000 @ log 1+2^-30 Q62
.word 0x80000000, 0x00000000 @ log 1+2^-31 Q62
.word 0x40000000, 0x00000000 @ log 1+2^-32 Q62
qfp_lib_end: