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List: openbsd-bugs
Subject: Re: Replace cos and avoid FPU trigonometry (was: tanf returns NaN for large inputs)
From: Mark Kettenis <mark.kettenis () xs4all ! nl>
Date: 2022-01-18 10:15:24
Message-ID: d3cbbc353a35694d () bloch ! sibelius ! xs4all ! nl
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> From: Greg Steuck <gnezdo@openbsd.org>
> Date: Mon, 10 Jan 2022 20:59:17 -0800
>
> Greg Steuck <gnezdo@openbsd.org> writes:
>
> > This failure can be reduced to a trivial program which does change
> > its behavior for the worse if s_cos.S is taken out:
> >
> > #include <math.h>
> > #include <stdio.h>
> >
> > int main(int a, char**b) {
> > double y = -0.34061437849088045332;
> > printf("cos(%lf)=%le delta=%e\n", y, cos(y), 0.94254960031831729956 - cos(y));
> > }
> >
> > In HEAD:
> >
> > cos(-0.340614)=9.425496e-01 delta=-1.110223e-16
> >
> > while with the patch below:
> >
> > cos(-0.340614)=9.425496e-01 delta=0.000000e+00
>
> As Daniel noted, I swapped the cases. The HEAD is at 0.0 delta whereas
> the patch used to make it worse.
>
> I went looking for why things are better on FreeBSD and they have a
> different (simpler) implementation of cos. I copied it over. Given the
> common provenance, I expect the copyright situation to be unambiguous.
I think you will also need the changes done in FreeBSD commit
4339c67c485f.
> With the two patches things look almost universally better in
> regress/libm. I attached both logs from amd64.
>
> Anybody has ideas for other tests that make sense to do? Maybe people
> can help me run regress on less common platforms?
>
> Thanks
> Greg
>
> >From a0b065bd3f5d48786f77f654dfb53cbf2617b0b3 Mon Sep 17 00:00:00 2001
> From: Greg Steuck <greg@nest.cx>
> Date: Mon, 10 Jan 2022 20:22:07 -0800
> Subject: [PATCH 1/2] Copy cos(3) software implementation from FreeBSD-13
>
> The result passes more tests from msun suite. In particular,
> testacc(cos, -0.34061437849088045332L, 0.94254960031831729956L,
> ALL_STD_EXCEPT, FE_INEXACT);
> matches instead of being 1e-16 off.
> ---
> lib/libm/src/k_cos.c | 45 ++++++++++++++++++--------------------------
> lib/libm/src/s_cos.c | 6 +++++-
> 2 files changed, 23 insertions(+), 28 deletions(-)
>
> diff --git a/lib/libm/src/k_cos.c b/lib/libm/src/k_cos.c
> index 8f3882b6a00..0839243e90c 100644
> --- a/lib/libm/src/k_cos.c
> +++ b/lib/libm/src/k_cos.c
> @@ -36,13 +36,17 @@
> * ~ cos(x) - x*y,
> * a correction term is necessary in cos(x) and hence
> * cos(x+y) = 1 - (x*x/2 - (r - x*y))
> - * For better accuracy when x > 0.3, let qx = |x|/4 with
> - * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
> - * Then
> - * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
> - * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
> - * magnitude of the latter is at least a quarter of x*x/2,
> - * thus, reducing the rounding error in the subtraction.
> + * For better accuracy, rearrange to
> + * cos(x+y) ~ w + (tmp + (r-x*y))
> + * where w = 1 - x*x/2 and tmp is a tiny correction term
> + * (1 - x*x/2 == w + tmp exactly in infinite precision).
> + * The exactness of w + tmp in infinite precision depends on w
> + * and tmp having the same precision as x. If they have extra
> + * precision due to compiler bugs, then the extra precision is
> + * only good provided it is retained in all terms of the final
> + * expression for cos(). Retention happens in all cases tested
> + * under FreeBSD, so don't pessimize things by forcibly clipping
> + * any extra precision in w.
> */
>
> #include "math.h"
> @@ -60,25 +64,12 @@ C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
> double
> __kernel_cos(double x, double y)
> {
> - double a,hz,z,r,qx;
> - int32_t ix;
> - GET_HIGH_WORD(ix,x);
> - ix &= 0x7fffffff; /* ix = |x|'s high word*/
> - if(ix<0x3e400000) { /* if x < 2**27 */
> - if(((int)x)==0) return one; /* generate inexact */
> - }
> + double hz,z,r,w;
> +
> z = x*x;
> - r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
> - if(ix < 0x3FD33333) /* if |x| < 0.3 */
> - return one - (0.5*z - (z*r - x*y));
> - else {
> - if(ix > 0x3fe90000) { /* x > 0.78125 */
> - qx = 0.28125;
> - } else {
> - INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */
> - }
> - hz = 0.5*z-qx;
> - a = one-qx;
> - return a - (hz - (z*r-x*y));
> - }
> + w = z*z;
> + r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
> + hz = 0.5*z;
> + w = one-hz;
> + return w + (((one-w)-hz) + (z*r-x*y));
> }
> diff --git a/lib/libm/src/s_cos.c b/lib/libm/src/s_cos.c
> index 8b923d5fe61..1406504e9ab 100644
> --- a/lib/libm/src/s_cos.c
> +++ b/lib/libm/src/s_cos.c
> @@ -57,7 +57,11 @@ cos(double x)
>
> /* |x| ~< pi/4 */
> ix &= 0x7fffffff;
> - if(ix <= 0x3fe921fb) return __kernel_cos(x,z);
> + if(ix <= 0x3fe921fb) {
> + if(ix<0x3e46a09e) /* if x < 2**-27 * sqrt(2) */
> + if(((int)x)==0) return 1.0; /* generate inexact */
> + return __kernel_cos(x,z);
> + }
>
> /* cos(Inf or NaN) is NaN */
> else if (ix>=0x7ff00000) return x-x;
> --
> 2.34.1
>
> >From 7461948f0e7ff41e5b3b9288d98edf13cb296742 Mon Sep 17 00:00:00 2001
> From: Greg Steuck <greg@nest.cx>
> Date: Sun, 9 Jan 2022 13:45:51 -0800
> Subject: [PATCH 2/2] Unplug assembly implementations of trig functions on x86
> platforms
>
> The same change was done by NetBSD some time back as:
>
> Disable x87 implementations of sin, cos, tan.
>
> The x87 hardware uses a bad approximation to pi for argument
> reduction, and consequently yields bad answers for inputs near pi or
> pi/2.
> ---
> lib/libm/Makefile | 9 ++++-----
> regress/lib/libm/msun/Makefile | 1 -
> 2 files changed, 4 insertions(+), 6 deletions(-)
>
> diff --git a/lib/libm/Makefile b/lib/libm/Makefile
> index 47cd94cac06..cd39bbe5c48 100644
> --- a/lib/libm/Makefile
> +++ b/lib/libm/Makefile
> @@ -22,11 +22,10 @@ ARCH_SRCS = e_acos.S e_asin.S e_atan2.S e_exp.S e_fmod.S e_log.S e_log10.S \
> e_sqrtl.S \
> invtrig.c \
> s_atan.S s_atanf.S s_ceil.S s_ceilf.S s_copysign.S s_copysignf.S \
> - s_cos.S s_cosf.S s_floor.S s_floorf.S \
> + s_floor.S s_floorf.S \
> s_log1p.S s_log1pf.S s_logb.S s_logbf.S \
> s_llrint.S s_llrintf.S s_lrint.S s_lrintf.S s_rint.S s_rintf.S\
> - s_scalbnf.S s_significand.S s_significandf.S \
> - s_sin.S s_sinf.S s_tan.S s_tanf.S
> + s_scalbnf.S s_significand.S s_significandf.S
> .elif (${MACHINE_ARCH} == "amd64")
> .PATH: ${.CURDIR}/arch/amd64
> CPPFLAGS+=-I${.CURDIR}/arch/amd64
> @@ -35,11 +34,11 @@ ARCH_SRCS = e_acos.S e_asin.S e_atan2.S e_exp.S e_fmod.S e_log.S e_log10.S \
> e_sqrtl.S \
> invtrig.c \
> s_atan.S s_atanf.S s_ceil.S s_ceilf.S s_copysign.S s_copysignf.S \
> - s_cos.S s_cosf.S s_floor.S s_floorf.S \
> + s_floor.S s_floorf.S \
> s_log1p.S s_log1pf.S s_logb.S s_logbf.S \
> s_llrint.S s_llrintf.S s_lrint.S s_lrintf.S \
> s_rint.S s_rintf.S s_scalbnf.S s_significand.S \
> - s_significandf.S s_sin.S s_sinf.S s_tan.S s_tanf.S
> + s_significandf.S
> .elif (${MACHINE_ARCH} == "hppa")
> .PATH: ${.CURDIR}/arch/hppa
> ARCH_SRCS = e_sqrt.c e_sqrtf.c e_remainder.c e_remainderf.c \
> diff --git a/regress/lib/libm/msun/Makefile b/regress/lib/libm/msun/Makefile
> index 1d6e9530c24..6f8bddf1831 100644
> --- a/regress/lib/libm/msun/Makefile
> +++ b/regress/lib/libm/msun/Makefile
> @@ -58,7 +58,6 @@ FAILING+= run-ctrig_test-1
> FAILING+= run-exponential_test-1
> FAILING+= run-invtrig_test-7
> FAILING+= run-next_test-{1,2,4}
> -FAILING+= run-trig_test-3
> . elif ${MACHINE} == arm64
> FAILING+= run-cexp_test-{1,7}
> FAILING+= run-ctrig_test-{1,5}
> --
> 2.34.1
>
>
> [2:text/plain Show Save:before (68kB)]
>
>
> [3:text/plain Show Save:after-all-soft (68kB)]
>
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