Remove slow paths from log

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.  A simple test over a few hundred
million values shows log is now 18% faster on average.

	* manual/probes.texi (slowlog): Delete documentation of removed probe.
	(slowlog_inexact): Likewise
	* sysdeps/ieee754/dbl-64/e_log.c (__ieee754_log): Remove slow paths.
	* sysdeps/ieee754/dbl-64/ulog.h: Remove unused declarations.

2 files changed, 16 insertions, 205 deletions

diff --git a/sysdeps/ieee754/dbl-64/e_log.c b/sysdeps/ieee754/dbl-64/e_log.c
index 6a18ebb904..2483dd8551 100644
--- 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 abs error is 6.095e-21 and min log (x) is 0.0295 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 36a31137b7..087b76e2ab 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