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# Re: [PATCH] Remove slow paths from log

```Joseph Myers wrote:

> > -  /* End stage I, case abs(x-1) < 0.03 */
> > -  if ((y = b + (c + b * E2)) == b + (c - b * E2))
> > -    return y;
>
>> +  return b + (c + b * E2);
>
> And in such a case, the analysis will imply that the b * E2 part of
> computing the result is not needed - it's actually computing one bound on
> the possible correctly rounded result - and you should just return b + c.

You're right - the multiply isn't strictly needed. I've removed it and added the
ULP computations based on the error bounds E1/E2:

+  /* Here b contains the high part of the result, and c the low part.
+     Maximum error is b * 2.334e-19, so accuracy is >61 bits.
+     Therefore max ULP error of b + c is ~0.502.  */
+  return b + c;

and the 2nd case:

+  /* Here A contains the high part of the result, and B the low part.
+     Maximum error is 6.095e-21 and min log (x) is 0.295 since x > 1.03.
+     Therefore max ULP error of A + B is ~0.502.  */
+  return A + B;

Wilco

Here is the updated version:

Remove the slow paths from log.  Like several other double precision math
functions, log is exactly rounded.  This is not required from math functions
and causes major overheads as it requires multiple fallbacks using higher
precision arithmetic if a result is close to 0.5ULP.  Ridiculous slowdowns
of up to 100000x have been reported when the highest precision path triggers.

Interestingly removing the slow paths makes hardly any difference in practice:
the worst case error is still ~0.502ULP, and exp(log(x)) shows identical results
before/after on many millions of random cases.  All GLIBC math tests pass on
AArch64 and x64 with no change in ULP error.

OK for commit?

ChangeLog:
2018-02-02  Wilco Dijkstra  <wdijkstr@arm.com>

* sysdeps/ieee754/dbl-64/e_log.c (__ieee754_log): Remove slow paths.
* sysdeps/ieee754/dbl-64/ulog.h: Remove unused declarations.
--

diff --git a/sysdeps/ieee754/dbl-64/e_log.c b/sysdeps/ieee754/dbl-64/e_log.c
--- a/sysdeps/ieee754/dbl-64/e_log.c
+++ b/sysdeps/ieee754/dbl-64/e_log.c
@@ -23,11 +23,10 @@
/*      FUNCTION:ulog                                                */
/*                                                                   */
/*      FILES NEEDED: dla.h endian.h mpa.h mydefs.h ulog.h           */
-/*                    mpexp.c mplog.c mpa.c                          */
/*                    ulog.tbl                                       */
/*                                                                   */
/* An ultimate log routine. Given an IEEE double machine number x    */
-/* it computes the correctly rounded (to nearest) value of log(x).   */
+/* it computes the rounded (to nearest) value of log(x).	     */
/* Assumption: Machine arithmetic operations are performed in        */
/* round to nearest mode of IEEE 754 standard.                       */
/*                                                                   */
@@ -40,34 +39,26 @@
#include "MathLib.h"
#include <math.h>
#include <math_private.h>
-#include <stap-probe.h>

#ifndef SECTION
# define SECTION
#endif

-void __mplog (mp_no *, mp_no *, int);
-
/*********************************************************************/
-/* An ultimate log routine. Given an IEEE double machine number x     */
-/* it computes the correctly rounded (to nearest) value of log(x).   */
+/* An ultimate log routine. Given an IEEE double machine number x    */
+/* it computes the rounded (to nearest) value of log(x).	     */
/*********************************************************************/
double
SECTION
__ieee754_log (double x)
{
-#define M 4
-  static const int pr[M] = { 8, 10, 18, 32 };
-  int i, j, n, ux, dx, p;
+  int i, j, n, ux, dx;
double dbl_n, u, p0, q, r0, w, nln2a, luai, lubi, lvaj, lvbj,
-	 sij, ssij, ttij, A, B, B0, y, y1, y2, polI, polII, sa, sb,
-	 t1, t2, t7, t8, t, ra, rb, ww,
-	 a0, aa0, s1, s2, ss2, s3, ss3, a1, aa1, a, aa, b, bb, c;
+	 sij, ssij, ttij, A, B, B0, polI, polII, t8, a, aa, b, bb, c;
#ifndef DLA_FMS
-  double t3, t4, t5, t6;
+  double t1, t2, t3, t4, t5;
#endif
number num;
-  mp_no mpx, mpy, mpy1, mpy2, mperr;

#include "ulog.tbl"
#include "ulog.h"
@@ -101,7 +92,7 @@ __ieee754_log (double x)
if (w == 0.0)
return 0.0;

-  /*--- Stage I, the case abs(x-1) < 0.03 */
+  /*--- The case abs(x-1) < 0.03 */

t8 = MHALF * w;
EMULV (t8, w, a, aa, t1, t2, t3, t4, t5);
@@ -118,50 +109,12 @@ __ieee754_log (double x)
polII *= w * w * w;
c = (aa + bb) + polII;

-  /* End stage I, case abs(x-1) < 0.03 */
-  if ((y = b + (c + b * E2)) == b + (c - b * E2))
-    return y;
-
-  /*--- Stage II, the case abs(x-1) < 0.03 */
-
-  a = d19.d + w * d20.d;
-  a = d18.d + w * a;
-  a = d17.d + w * a;
-  a = d16.d + w * a;
-  a = d15.d + w * a;
-  a = d14.d + w * a;
-  a = d13.d + w * a;
-  a = d12.d + w * a;
-  a = d11.d + w * a;
-
-  EMULV (w, a, s2, ss2, t1, t2, t3, t4, t5);
-  ADD2 (d10.d, dd10.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (d9.d, dd9.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (d8.d, dd8.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (d7.d, dd7.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (d6.d, dd6.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (d5.d, dd5.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (d4.d, dd4.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (d3.d, dd3.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (d2.d, dd2.d, s2, ss2, s3, ss3, t1, t2);
-  MUL2 (w, 0, s3, ss3, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  MUL2 (w, 0, s2, ss2, s3, ss3, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (w, 0, s3, ss3, b, bb, t1, t2);
+  /* Here b contains the high part of the result, and c the low part.
+     Maximum error is b * 2.334e-19, so accuracy is >61 bits.
+     Therefore max ULP error of b + c is ~0.502.  */
+  return b + c;

-  /* End stage II, case abs(x-1) < 0.03 */
-  if ((y = b + (bb + b * E4)) == b + (bb - b * E4))
-    return y;
-  goto stage_n;
-
-  /*--- Stage I, the case abs(x-1) > 0.03 */
+  /*--- The case abs(x-1) > 0.03 */
case_03:

/* Find n,u such that x = u*2**n,   1/sqrt(2) < u < sqrt(2)  */
@@ -203,58 +156,10 @@ case_03:
B0 = (((lubi + lvbj) + ssij) + ttij) + dbl_n * LN2B;
B = polI + B0;

-  /* End stage I, case abs(x-1) >= 0.03 */
-  if ((y = A + (B + E1)) == A + (B - E1))
-    return y;
-
-
-  /*--- Stage II, the case abs(x-1) > 0.03 */
-
-  /* Improve the accuracy of r0 */
-  EMULV (p0, r0, sa, sb, t1, t2, t3, t4, t5);
-  t = r0 * ((1 - sa) - sb);
-  EADD (r0, t, ra, rb);
-
-  /* Compute w */
-  MUL2 (q, 0, ra, rb, w, ww, t1, t2, t3, t4, t5, t6, t7, t8);
-
-  EADD (A, B0, a0, aa0);
-
-  /* Evaluate polynomial III */
-  s1 = (c3.d + (c4.d + c5.d * w) * w) * w;
-  EADD (c2.d, s1, s2, ss2);
-  MUL2 (s2, ss2, w, ww, s3, ss3, t1, t2, t3, t4, t5, t6, t7, t8);
-  MUL2 (s3, ss3, w, ww, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
-  ADD2 (s2, ss2, w, ww, s3, ss3, t1, t2);
-  ADD2 (s3, ss3, a0, aa0, a1, aa1, t1, t2);
-
-  /* End stage II, case abs(x-1) >= 0.03 */
-  if ((y = a1 + (aa1 + E3)) == a1 + (aa1 - E3))
-    return y;
-
-
-  /* Final stages. Use multi-precision arithmetic. */
-stage_n:
-
-  for (i = 0; i < M; i++)
-    {
-      p = pr[i];
-      __dbl_mp (x, &mpx, p);
-      __dbl_mp (y, &mpy, p);
-      __mplog (&mpx, &mpy, p);
-      __dbl_mp (e[i].d, &mperr, p);
-      __add (&mpy, &mperr, &mpy1, p);
-      __sub (&mpy, &mperr, &mpy2, p);
-      __mp_dbl (&mpy1, &y1, p);
-      __mp_dbl (&mpy2, &y2, p);
-      if (y1 == y2)
-	{
-	  LIBC_PROBE (slowlog, 3, &p, &x, &y1);
-	  return y1;
-	}
-    }
-  LIBC_PROBE (slowlog_inexact, 3, &p, &x, &y1);
-  return y1;
+  /* Here A contains the high part of the result, and B the low part.
+     Maximum error is 6.095e-21 and min log (x) is 0.295 since x > 1.03.
+     Therefore max ULP error of A + B is ~0.502.  */
+  return A + B;
}

#ifndef __ieee754_log
diff --git a/sysdeps/ieee754/dbl-64/ulog.h b/sysdeps/ieee754/dbl-64/ulog.h
index 36a31137b759f604fba611d68a60efc90dc8d20d..087b76e2abaa7e9530c7195c48112db3c851e86b 100644
--- a/sysdeps/ieee754/dbl-64/ulog.h
+++ b/sysdeps/ieee754/dbl-64/ulog.h
@@ -42,43 +42,6 @@
/**/ b6             = {{0x3fbc71c5, 0x25db58ac} }, /*  0.111... */
/**/ b7             = {{0xbfb9a4ac, 0x11a2a61c} }, /* -0.100... */
/**/ b8             = {{0x3fb75077, 0x0df2b591} }, /*  0.091... */
-  /* polynomial III */
-#if 0
-/**/ c1             = {{0x3ff00000, 0x00000000} }, /*  1        */
-#endif
-/**/ c2             = {{0xbfe00000, 0x00000000} }, /* -1/2      */
-/**/ c3             = {{0x3fd55555, 0x55555555} }, /*  1/3      */
-/**/ c4             = {{0xbfd00000, 0x00000000} }, /* -1/4      */
-/**/ c5             = {{0x3fc99999, 0x9999999a} }, /*  1/5      */
-  /* polynomial IV */
-/**/ d2             = {{0xbfe00000, 0x00000000} }, /* -1/2      */
-/**/ dd2            = {{0x00000000, 0x00000000} }, /* -1/2-d2   */
-/**/ d3             = {{0x3fd55555, 0x55555555} }, /*  1/3      */
-/**/ dd3            = {{0x3c755555, 0x55555555} }, /*  1/3-d3   */
-/**/ d4             = {{0xbfd00000, 0x00000000} }, /* -1/4      */
-/**/ dd4            = {{0x00000000, 0x00000000} }, /* -1/4-d4   */
-/**/ d5             = {{0x3fc99999, 0x9999999a} }, /*  1/5      */
-/**/ dd5            = {{0xbc699999, 0x9999999a} }, /*  1/5-d5   */
-/**/ d6             = {{0xbfc55555, 0x55555555} }, /* -1/6      */
-/**/ dd6            = {{0xbc655555, 0x55555555} }, /* -1/6-d6   */
-/**/ d7             = {{0x3fc24924, 0x92492492} }, /*  1/7      */
-/**/ dd7            = {{0x3c624924, 0x92492492} }, /*  1/7-d7   */
-/**/ d8             = {{0xbfc00000, 0x00000000} }, /* -1/8      */
-/**/ dd8            = {{0x00000000, 0x00000000} }, /* -1/8-d8   */
-/**/ d9             = {{0x3fbc71c7, 0x1c71c71c} }, /*  1/9      */
-/**/ dd9            = {{0x3c5c71c7, 0x1c71c71c} }, /*  1/9-d9   */
-/**/ d10            = {{0xbfb99999, 0x9999999a} }, /* -1/10     */
-/**/ dd10           = {{0x3c599999, 0x9999999a} }, /* -1/10-d10 */
-/**/ d11            = {{0x3fb745d1, 0x745d1746} }, /*  1/11     */
-/**/ d12            = {{0xbfb55555, 0x55555555} }, /* -1/12     */
-/**/ d13            = {{0x3fb3b13b, 0x13b13b14} }, /*  1/13     */
-/**/ d14            = {{0xbfb24924, 0x92492492} }, /* -1/14     */
-/**/ d15            = {{0x3fb11111, 0x11111111} }, /*  1/15     */
-/**/ d16            = {{0xbfb00000, 0x00000000} }, /* -1/16     */
-/**/ d17            = {{0x3fae1e1e, 0x1e1e1e1e} }, /*  1/17     */
-/**/ d18            = {{0xbfac71c7, 0x1c71c71c} }, /* -1/18     */
-/**/ d19            = {{0x3faaf286, 0xbca1af28} }, /*  1/19     */
-/**/ d20            = {{0xbfa99999, 0x9999999a} }, /* -1/20     */
/* constants    */
/**/ sqrt_2         = {{0x3ff6a09e, 0x667f3bcc} }, /* sqrt(2)   */
/**/ h1             = {{0x3fd2e000, 0x00000000} }, /* 151/2**9  */
@@ -87,14 +50,6 @@
/**/ delv           = {{0x3ef00000, 0x00000000} }, /* 1/2**16   */
/**/ ln2a           = {{0x3fe62e42, 0xfefa3800} }, /* ln(2) 43 bits */
/**/ ln2b           = {{0x3d2ef357, 0x93c76730} }, /* ln(2)-ln2a    */
-/**/ e1             = {{0x3bbcc868, 0x00000000} }, /* 6.095e-21     */
-/**/ e2             = {{0x3c1138ce, 0x00000000} }, /* 2.334e-19     */
-/**/ e3             = {{0x3aa1565d, 0x00000000} }, /* 2.801e-26     */
-/**/ e4             = {{0x39809d88, 0x00000000} }, /* 1.024e-31     */
-/**/ e[M]           ={{{0x37da223a, 0x00000000} }, /* 1.2e-39       */
-/**/                  {{0x35c851c4, 0x00000000} }, /* 1.3e-49       */
-/**/                  {{0x2ab85e51, 0x00000000} }, /* 6.8e-103      */
-/**/                  {{0x17383827, 0x00000000} }},/* 8.1e-197      */
/**/ two54          = {{0x43500000, 0x00000000} }, /* 2**54         */
/**/ u03            = {{0x3f9eb851, 0xeb851eb8} }; /* 0.03          */

@@ -114,43 +69,6 @@
/**/ b6             = {{0x25db58ac, 0x3fbc71c5} }, /*  0.111... */
/**/ b7             = {{0x11a2a61c, 0xbfb9a4ac} }, /* -0.100... */
/**/ b8             = {{0x0df2b591, 0x3fb75077} }, /*  0.091... */
-  /* polynomial III */
-#if 0
-/**/ c1             = {{0x00000000, 0x3ff00000} }, /*  1        */
-#endif
-/**/ c2             = {{0x00000000, 0xbfe00000} }, /* -1/2      */
-/**/ c3             = {{0x55555555, 0x3fd55555} }, /*  1/3      */
-/**/ c4             = {{0x00000000, 0xbfd00000} }, /* -1/4      */
-/**/ c5             = {{0x9999999a, 0x3fc99999} }, /*  1/5      */
-  /* polynomial IV */
-/**/ d2             = {{0x00000000, 0xbfe00000} }, /* -1/2      */
-/**/ dd2            = {{0x00000000, 0x00000000} }, /* -1/2-d2   */
-/**/ d3             = {{0x55555555, 0x3fd55555} }, /*  1/3      */
-/**/ dd3            = {{0x55555555, 0x3c755555} }, /*  1/3-d3   */
-/**/ d4             = {{0x00000000, 0xbfd00000} }, /* -1/4      */
-/**/ dd4            = {{0x00000000, 0x00000000} }, /* -1/4-d4   */
-/**/ d5             = {{0x9999999a, 0x3fc99999} }, /*  1/5      */
-/**/ dd5            = {{0x9999999a, 0xbc699999} }, /*  1/5-d5   */
-/**/ d6             = {{0x55555555, 0xbfc55555} }, /* -1/6      */
-/**/ dd6            = {{0x55555555, 0xbc655555} }, /* -1/6-d6   */
-/**/ d7             = {{0x92492492, 0x3fc24924} }, /*  1/7      */
-/**/ dd7            = {{0x92492492, 0x3c624924} }, /*  1/7-d7   */
-/**/ d8             = {{0x00000000, 0xbfc00000} }, /* -1/8      */
-/**/ dd8            = {{0x00000000, 0x00000000} }, /* -1/8-d8   */
-/**/ d9             = {{0x1c71c71c, 0x3fbc71c7} }, /*  1/9      */
-/**/ dd9            = {{0x1c71c71c, 0x3c5c71c7} }, /*  1/9-d9   */
-/**/ d10            = {{0x9999999a, 0xbfb99999} }, /* -1/10     */
-/**/ dd10           = {{0x9999999a, 0x3c599999} }, /* -1/10-d10 */
-/**/ d11            = {{0x745d1746, 0x3fb745d1} }, /*  1/11     */
-/**/ d12            = {{0x55555555, 0xbfb55555} }, /* -1/12     */
-/**/ d13            = {{0x13b13b14, 0x3fb3b13b} }, /*  1/13     */
-/**/ d14            = {{0x92492492, 0xbfb24924} }, /* -1/14     */
-/**/ d15            = {{0x11111111, 0x3fb11111} }, /*  1/15     */
-/**/ d16            = {{0x00000000, 0xbfb00000} }, /* -1/16     */
-/**/ d17            = {{0x1e1e1e1e, 0x3fae1e1e} }, /*  1/17     */
-/**/ d18            = {{0x1c71c71c, 0xbfac71c7} }, /* -1/18     */
-/**/ d19            = {{0xbca1af28, 0x3faaf286} }, /*  1/19     */
-/**/ d20            = {{0x9999999a, 0xbfa99999} }, /* -1/20     */
/* constants    */
/**/ sqrt_2         = {{0x667f3bcc, 0x3ff6a09e} }, /* sqrt(2)   */
/**/ h1             = {{0x00000000, 0x3fd2e000} }, /* 151/2**9  */
@@ -159,14 +77,6 @@
/**/ delv           = {{0x00000000, 0x3ef00000} }, /* 1/2**16   */
/**/ ln2a           = {{0xfefa3800, 0x3fe62e42} }, /* ln(2) 43 bits */
/**/ ln2b           = {{0x93c76730, 0x3d2ef357} }, /* ln(2)-ln2a    */
-/**/ e1             = {{0x00000000, 0x3bbcc868} }, /* 6.095e-21     */
-/**/ e2             = {{0x00000000, 0x3c1138ce} }, /* 2.334e-19     */
-/**/ e3             = {{0x00000000, 0x3aa1565d} }, /* 2.801e-26     */
-/**/ e4             = {{0x00000000, 0x39809d88} }, /* 1.024e-31     */
-/**/ e[M]           ={{{0x00000000, 0x37da223a} }, /* 1.2e-39       */
-/**/                  {{0x00000000, 0x35c851c4} }, /* 1.3e-49       */
-/**/                  {{0x00000000, 0x2ab85e51} }, /* 6.8e-103      */
-/**/                  {{0x00000000, 0x17383827} }},/* 8.1e-197      */
/**/ two54          = {{0x00000000, 0x43500000} }, /* 2**54         */
/**/ u03            = {{0xeb851eb8, 0x3f9eb851} }; /* 0.03          */

@@ -178,10 +88,6 @@
#define  DEL_V     delv.d
#define  LN2A      ln2a.d
#define  LN2B      ln2b.d
-#define  E1        e1.d
-#define  E2        e2.d
-#define  E3        e3.d
-#define  E4        e4.d
#define  U03       u03.d

#endif

```

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