@ 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_fexp
.global qfp_fln
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
.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 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
.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
.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
qfp_lib_end: