Patchwork Remove slow paths from log

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Submitter Wilco Dijkstra
Date Feb. 2, 2018, 5:44 p.m.
Message ID <DB6PR0801MB2053DB22BAA9B3C15011BBBD83F90@DB6PR0801MB2053.eurprd08.prod.outlook.com>
Download mbox | patch
Permalink /patch/25777/
State New
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Comments

Wilco Dijkstra - Feb. 2, 2018, 5:44 p.m.
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 ~0.5ULP, 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.
--
Joseph Myers - Feb. 2, 2018, 5:56 p.m.
On Fri, 2 Feb 2018, Wilco Dijkstra wrote:

> Interestingly removing the slow paths makes hardly any difference in practice:
> the worst case error is still ~0.5ULP, 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.

Could you please give the error analysis that justifies that the worst 
case error is still ~0.5ulp?

I don't expect a direct error analysis of the fast case code, though you 
can do one if you want.  Rather, in such cases, an indirect justification 
of the following form (which supposes that the existing code does have an 
error bound of 0.5ulp) is probably a lot easier to produce:

The fast case produces a result of the form HIGH + LOW, with error bound 
E, and checks that HIGH + (LOW + E) and HIGH + (LOW - E) round to the same 
value to nearest - if they don't, it uses extra precision.  That is, on 
the assumption that the existing code is correct, E is an error bound for 
HIGH + LOW.  And assuming that E is less than 0.5ulp of the correct 
result, that means we don't actually need the error calculation, and can 
get sufficiently good results just from the fast path.

I suspect that analyses of this form will allow eliminating most if not 
all of the slow paths.  But it's necessary in each case where removal of a 
slow path is produced to say what E is, what the minimum magnitude of the 
correct result is, and so how small E must be in terms of the correct 
result, to justify that the slow path can be removed.

> -  /* 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.

Patch

diff --git a/sysdeps/ieee754/dbl-64/e_log.c b/sysdeps/ieee754/dbl-64/e_log.c
index 6a18ebb904fc42a69ed72d79f6db646addf46054..cd8ac3534d642bfb936483696c1ff7f880319538 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 +109,9 @@  __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);
+  return b + (c + b * E2);
 
-  /* 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 +153,7 @@  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;
+  return A + (B + E1);
 }
 
 #ifndef __ieee754_log
diff --git a/sysdeps/ieee754/dbl-64/ulog.h b/sysdeps/ieee754/dbl-64/ulog.h
index 36a31137b759f604fba611d68a60efc90dc8d20d..b9bb8ab046ed1b10bf1bf8a7b66c820d9d6a39cb 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  */
@@ -89,12 +52,6 @@ 
 /**/ 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 +71,6 @@ 
 /**/ 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  */
@@ -161,12 +81,6 @@ 
 /**/ 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          */