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

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"
 
-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 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 {
-			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;
 }

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, [email protected].
- */
-/*
- * ====================================================
- * 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"
 
-static const float
-tiny = 1.0e-30,
-huge = 1.0e30;
-
 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 {
-			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;
 }

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
@@ -41,43 +6,40 @@ long double tanhl(long double x)
 	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 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 {
-			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 {
-		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