Re: Replace cos and avoid FPU trigonometry (was: tanf returns NaN for large inputs)
> From: Greg Steuck
> Date: Mon, 10 Jan 2022 20:59:17 -0800
>
> Greg Steuck 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
> > #include
> >
> > 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=0.00e+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
> 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 &= 0x7fff; /* ix = |x|'s high word*/
> - if(ix<0x3e40) { /* 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 < 0x3FD3) /* if |x| < 0.3 */
> - return one - (0.5*z - (z*r - x*y));
> - else {
> - if(ix > 0x3fe9) { /* x > 0.78125 */
> - qx = 0.28125;
> - } else {
> - INSERT_WORDS(qx,ix-0x0020,0); /* 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 &= 0x7fff;
> - 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);
> + }
Replace cos and avoid FPU trigonometry (was: tanf returns NaN for large inputs)
Greg Steuck 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
> #include
>
> 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=0.00e+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.
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
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 &= 0x7fff; /* ix = |x|'s high word*/
- if(ix<0x3e40) { /* 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 < 0x3FD3) /* if |x| < 0.3 */
- return one - (0.5*z - (z*r - x*y));
- else {
- if(ix > 0x3fe9) { /* x > 0.78125 */
- qx = 0.28125;
- } else {
- INSERT_WORDS(qx,ix-0x0020,0); /* 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 &= 0x7fff;
- 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>=0x7ff0) return x-x;
--
2.34.1
>From 7461948f0e7ff41e5b3b9288d98edf13cb296742 Mon Sep 17 00:00:00 2001
From: Greg Steuck
Date: Sun, 9 Jan 2022 13:45:51 -0800
Subject: [PATCH 2/2] Unplug assembly implementations of
