math: tanh.c cleanup similar to sinh, cosh

comments are kept in the double version of the function

compared to fdlibm/freebsd we partition the domain into one
more part and select different threshold points:
now the [log(5/3)/2,log(3)/2] and [log(3)/2,inf] domains
should have <1.5ulp error
(so only the last bit may be wrong, assuming good exp, expm1)

(note that log(3)/2 and log(5/3)/2 are the points where tanh
changes resolution: tanh(log(3)/2)=0.5, tanh(log(5/3)/2)=0.25)

for some x < log(5/3)/2 (~=0.2554) the error can be >1.5ulp
but it should be <2ulp
(the freebsd code had some >2ulp errors in [0.255,1])

even with the extra logic the new code produces smaller
object files

diff --git a/src/math/tanh.c b/src/math/tanh.c
index 2113864..0e766c5 100644 (file)
--- a/src/math/tanh.c
+++ b/src/math/tanh.c
@@ -1,73 +1,41 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/s_tanh.c */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-/* Tanh(x)
- * Return the Hyperbolic Tangent of x
- *
- * Method :
- *                                     x    -x
- *                                    e  - e
- *      0. tanh(x) is defined to be -----------
- *                                     x    -x
- *                                    e  + e
- *      1. reduce x to non-negative by tanh(-x) = -tanh(x).
- *      2.  0      <= x <  2**-28 : tanh(x) := x with inexact if x != 0
- *                                              -t
- *          2**-28 <= x <  1      : tanh(x) := -----; t = expm1(-2x)
- *                                             t + 2
- *                                                   2
- *          1      <= x <  22     : tanh(x) := 1 - -----; t = expm1(2x)
- *                                                 t + 2
- *          22     <= x <= INF    : tanh(x) := 1.
- *
- * Special cases:
- *      tanh(NaN) is NaN;
- *      only tanh(0)=0 is exact for finite argument.
- */
-

 #include "libm.h"
 
 #include "libm.h"
 

-static const double tiny = 1.0e-300, huge = 1.0e300;
-
+/* tanh(x) = (exp(x) - exp(-x))/(exp(x) + exp(-x))
+ *         = (exp(2*x) - 1)/(exp(2*x) - 1 + 2)
+ *         = (1 - exp(-2*x))/(exp(-2*x) - 1 + 2)
+ */

 double tanh(double x)
 {
 double tanh(double x)
 {

-       double t,z;
-       int32_t jx,ix;
-
-       GET_HIGH_WORD(jx, x);
-       ix = jx & 0x7fffffff;
+       union {double f; uint64_t i;} u = {.f = x};
+       uint32_t w;
+       int sign;
+       double t;

 
 

-       /* x is INF or NaN */
-       if (ix >= 0x7ff00000) {
-               if (jx >= 0)
-                       return 1.0f/x + 1.0f;  /* tanh(+-inf)=+-1 */
-               else
-                       return 1.0f/x - 1.0f;  /* tanh(NaN) = NaN */
-       }
+       /* x = |x| */
+       sign = u.i >> 63;
+       u.i &= (uint64_t)-1/2;
+       x = u.f;
+       w = u.i >> 32;

 
 

-       if (ix < 0x40360000) {  /* |x| < 22 */
-               if (ix < 0x3e300000) {  /* |x| < 2**-28 */
-                       /* tanh(tiny) = tiny with inexact */
-                       if (huge+x > 1.0f)
-                               return x;
-               }
-               if (ix >= 0x3ff00000) {  /* |x| >= 1  */
-                       t = expm1(2.0f*fabs(x));
-                       z = 1.0f - 2.0f/(t+2.0f);
+       if (w > 0x3fe193ea) {
+               /* |x| > log(3)/2 ~= 0.5493 or nan */
+               if (w > 0x40340000) {
+                       /* |x| > 20 or nan */
+                       /* note: this branch avoids raising overflow */
+                       /* raise inexact if x!=+-inf and handle nan */
+                       t = 1 + 0/(x + 0x1p-120f);

                } else {
                } else {

-                       t = expm1(-2.0f*fabs(x));
-                       z= -t/(t+2.0f);
+                       t = expm1(2*x);
+                       t = 1 - 2/(t+2);

                }
                }

-       } else {  /* |x| >= 22, return +-1 */
-               z = 1.0f - tiny;  /* raise inexact */
+       } else if (w > 0x3fd058ae) {
+               /* |x| > log(5/3)/2 ~= 0.2554 */
+               t = expm1(2*x);
+               t = t/(t+2);
+       } else {
+               /* |x| is small, up to 2ulp error in [0.1,0.2554] */
+               t = expm1(-2*x);
+               t = -t/(t+2);

        }
        }

-       return jx >= 0 ? z : -z;
+       return sign ? -t : t;

 }
 }

diff --git a/src/math/tanhf.c b/src/math/tanhf.c
index 7cb459d..8099ec3 100644 (file)
--- a/src/math/tanhf.c
+++ b/src/math/tanhf.c
@@ -1,55 +1,35 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/s_tanhf.c */
-/*
- * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
- */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-

 #include "libm.h"
 
 #include "libm.h"
 

-static const float
-tiny = 1.0e-30,
-huge = 1.0e30;
-

 float tanhf(float x)
 {
 float tanhf(float x)
 {

-       float t,z;
-       int32_t jx,ix;
+       union {float f; uint32_t i;} u = {.f = x};
+       uint32_t w;
+       int sign;
+       float t;

 
 

-       GET_FLOAT_WORD(jx, x);
-       ix = jx & 0x7fffffff;
+       /* x = |x| */
+       sign = u.i >> 31;
+       u.i &= 0x7fffffff;
+       x = u.f;
+       w = u.i;

 
 

-       /* x is INF or NaN */
-       if(ix >= 0x7f800000) {
-               if (jx >= 0)
-                       return 1.0f/x + 1.0f;  /* tanh(+-inf)=+-1 */
-               else
-                       return 1.0f/x - 1.0f;  /* tanh(NaN) = NaN */
-       }
-
-       if (ix < 0x41100000) {  /* |x| < 9 */
-               if (ix < 0x39800000) {  /* |x| < 2**-12 */
-                       /* tanh(tiny) = tiny with inexact */
-                       if (huge+x > 1.0f)
-                               return x;
-               }
-               if (ix >= 0x3f800000) {  /* |x|>=1  */
-                       t = expm1f(2.0f*fabsf(x));
-                       z = 1.0f - 2.0f/(t+2.0f);
+       if (w > 0x3f0c9f54) {
+               /* |x| > log(3)/2 ~= 0.5493 or nan */
+               if (w > 0x41200000) {
+                       /* |x| > 10 */
+                       t = 1 + 0/(x + 0x1p-120f);

                } else {
                } else {

-                       t = expm1f(-2.0f*fabsf(x));
-                       z = -t/(t+2.0f);
+                       t = expm1f(2*x);
+                       t = 1 - 2/(t+2);

                }
                }

-       } else {  /* |x| >= 9, return +-1 */
-               z = 1.0f - tiny;  /* raise inexact */
+       } else if (w > 0x3e82c578) {
+               /* |x| > log(5/3)/2 ~= 0.2554 */
+               t = expm1f(2*x);
+               t = t/(t+2);
+       } else {
+               /* |x| is small */
+               t = expm1f(-2*x);
+               t = -t/(t+2);

        }
        }

-       return jx >= 0 ? z : -z;
+       return sign ? -t : t;

 }
 }

diff --git a/src/math/tanhl.c b/src/math/tanhl.c
index 92efb20..66559e9 100644 (file)
--- a/src/math/tanhl.c
+++ b/src/math/tanhl.c
@@ -1,38 +1,3 @@
-/* origin: OpenBSD /usr/src/lib/libm/src/ld80/s_tanhl.c */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-/* tanhl(x)
- * Return the Hyperbolic Tangent of x
- *
- * Method :
- *                                      x    -x
- *                                     e  - e
- *      0. tanhl(x) is defined to be -----------
- *                                      x    -x
- *                                     e  + e
- *      1. reduce x to non-negative by tanhl(-x) = -tanhl(x).
- *      2.  0      <= x <= 2**-55 : tanhl(x) := x*(one+x)
- *                                               -t
- *          2**-55 <  x <=  1     : tanhl(x) := -----; t = expm1l(-2x)
- *                                              t + 2
- *                                                    2
- *          1      <= x <=  23.0  : tanhl(x) := 1-  ----- ; t=expm1l(2x)
- *                                                  t + 2
- *          23.0   <  x <= INF    : tanhl(x) := 1.
- *
- * Special cases:
- *      tanhl(NaN) is NaN;
- *      only tanhl(0)=0 is exact for finite argument.
- */
-

 #include "libm.h"
 
 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
 #include "libm.h"
 
 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024


@@ -41,43 +6,40 @@ long double tanhl(long double x)
        return tanh(x);
 }
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
        return tanh(x);
 }
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384

-static const long double tiny = 1.0e-4900L;
-

 long double tanhl(long double x)
 {
 long double tanhl(long double x)
 {

-       long double t,z;
-       int32_t se;
-       uint32_t jj0,jj1,ix;
+       union {
+               long double f;
+               struct{uint64_t m; uint16_t se; uint16_t pad;} i;
+       } u = {.f = x};
+       unsigned ex = u.i.se & 0x7fff;
+       unsigned sign = u.i.se & 0x8000;
+       uint32_t w;
+       long double t;

 
 

-       /* High word of |x|. */
-       GET_LDOUBLE_WORDS(se, jj0, jj1, x);
-       ix = se & 0x7fff;
-
-       /* x is INF or NaN */
-       if (ix == 0x7fff) {
-               /* for NaN it's not important which branch: tanhl(NaN) = NaN */
-               if (se & 0x8000)
-                       return 1.0/x-1.0;  /* tanhl(-inf)= -1; */
-               return 1.0/x+1.0;          /* tanhl(+inf)= +1 */
-       }
+       /* x = |x| */
+       u.i.se = ex;
+       x = u.f;
+       w = u.i.m >> 32;

 
 

-       /* |x| < 23 */
-       if (ix < 0x4003 || (ix == 0x4003 && jj0 < 0xb8000000u)) {
-               if ((ix|jj0|jj1) == 0) /* x == +- 0 */
-                       return x;
-               if (ix < 0x3fc8)       /* |x| < 2**-55 */
-                       return x*(1.0+tiny);  /* tanh(small) = small */
-               if (ix >= 0x3fff) {    /* |x| >= 1  */
-                       t = expm1l(2.0*fabsl(x));
-                       z = 1.0 - 2.0/(t+2.0);
+       if (ex > 0x3ffe || (ex == 0x3ffe && w > 0x8c9f53d5)) {
+               /* |x| > log(3)/2 ~= 0.5493 or nan */
+               if (ex >= 0x3fff+5) {
+                       /* |x| >= 32 */
+                       t = 1 + 0/(x + 0x1p-120f);

                } else {
                } else {

-                       t = expm1l(-2.0*fabsl(x));
-                       z = -t/(t+2.0);
+                       t = expm1l(2*x);
+                       t = 1 - 2/(t+2);

                }
                }

-       /* |x| > 23, return +-1 */
+       } else if (ex > 0x3ffd || (ex == 0x3ffd && w > 0x82c577d4)) {
+               /* |x| > log(5/3)/2 ~= 0.2554 */
+               t = expm1l(2*x);
+               t = t/(t+2);

        } else {
        } else {

-               z = 1.0 - tiny;  /* raise inexact flag */
+               /* |x| is small */
+               t = expm1l(-2*x);
+               t = -t/(t+2);

        }
        }

-       return se & 0x8000 ? -z : z;
+       return sign ? -t : t;

 }
 #endif
 }
 #endif