math: finished cosh.c cleanup

changed the algorithm: large input is not special cased
(when exp(-x) is small compared to exp(x))
and the threshold values are reevaluated
(fdlibm code had a log(2)/2 cutoff for which i could not find
justification, log(2) seems to be a better threshold and this
was verified empirically)

the new code is simpler, makes smaller binaries and should be
faster for common cases

the old comments were removed as they are no longer true for the
new algorithm and the fdlibm copyright was dropped as well
because there is no common code or idea with the original anymore
except for trivial ones.

src/math/cosh.c

-/* origin: FreeBSD /usr/src/lib/msun/src/e_cosh.c */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunSoft, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-/* cosh(x)
- * Method :
- * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
- *      1. Replace x by |x| (cosh(x) = cosh(-x)).
- *      2.
- *                                                      [ exp(x) - 1 ]^2
- *          0        <= x <= ln2/2  :  cosh(x) := 1 + -------------------
- *                                                         2*exp(x)
- *
- *                                                exp(x) +  1/exp(x)
- *          ln2/2    <= x <= 22     :  cosh(x) := -------------------
- *                                                        2
- *          22       <= x <= lnovft :  cosh(x) := exp(x)/2
- *          lnovft   <= x <= ln2ovft:  cosh(x) := exp(x/2)/2 * exp(x/2)
- *          ln2ovft  <  x           :  cosh(x) := inf (overflow)
- *
- * Special cases:
- *      cosh(x) is |x| if x is +INF, -INF, or NaN.
- *      only cosh(0)=1 is exact for finite x.
- */
-
 #include "libm.h"
 
+/* cosh(x) = (exp(x) + 1/exp(x))/2
+ *         = 1 + 0.5*(exp(x)-1)*(exp(x)-1)/exp(x)
+ *         = 1 + x*x/2 + o(x^4)
+ */
 double cosh(double x)
 {
 	union {double f; uint64_t i;} u = {.f = x};
-	uint32_t ix;
+	uint32_t w;
 	double t;
 
 	/* |x| */
 	u.i &= (uint64_t)-1/2;
 	x = u.f;
-	ix = u.i >> 32;
+	w = u.i >> 32;
 
-	/* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
-	if (ix < 0x3fd62e43) {
-		t = expm1(x);
-		if (ix < 0x3c800000)
+	/* |x| < log(2) */
+	if (w < 0x3fe62e42) {
+		if (w < 0x3ff00000 - (26<<20)) {
+			/* raise inexact if x!=0 */
+			FORCE_EVAL(x + 0x1p120f);
 			return 1;
+		}
+		t = expm1(x);
 		return 1 + t*t/(2*(1+t));
 	}
 
-	/* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|))/2; */
-	if (ix < 0x40360000) {
+	/* |x| < log(DBL_MAX) */
+	if (w < 0x40862e42) {
 		t = exp(x);
-		return 0.5*t + 0.5/t;
+		/* note: if x>log(0x1p26) then the 1/t is not needed */
+		return 0.5*(t + 1/t);
 	}
 
-	/* |x| in [22, log(maxdouble)] return 0.5*exp(|x|) */
-	if (ix < 0x40862e42)
-		return 0.5*exp(x);
-
-	/* |x| in [log(maxdouble), overflowthresold] */
-	if (ix <= 0x408633ce)
-		return __expo2(x);
-
-	/* overflow (or nan) */
-	x *= 0x1p1023;
-	return x;
+	/* |x| > log(DBL_MAX) or nan */
+	/* note: the result is stored to handle overflow */
+	t = __expo2(x);
+	return t;
 }

src/math/coshf.c

-/* origin: FreeBSD /usr/src/lib/msun/src/e_coshf.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"
 
 float coshf(float x)
 {
 	union {float f; uint32_t i;} u = {.f = x};
-	uint32_t ix;
+	uint32_t w;
 	float t;
 
 	/* |x| */
 	u.i &= 0x7fffffff;
 	x = u.f;
-	ix = u.i;
+	w = u.i;
 
-	/* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
-	if (ix < 0x3eb17218) {
-		t = expm1f(x);
-		if (ix < 0x39800000)
+	/* |x| < log(2) */
+	if (w < 0x3f317217) {
+		if (w < 0x3f800000 - (12<<23)) {
+			FORCE_EVAL(x + 0x1p120f);
 			return 1;
+		}
+		t = expm1f(x);
 		return 1 + t*t/(2*(1+t));
 	}
 
-	/* |x| in [0.5*ln2,9], return (exp(|x|)+1/exp(|x|))/2; */
-	if (ix < 0x41100000) {
+	/* |x| < log(FLT_MAX) */
+	if (w < 0x42b17217) {
 		t = expf(x);
-		return 0.5f*t + 0.5f/t;
+		return 0.5f*(t + 1/t);
 	}
 
-	/* |x| in [9, log(maxfloat)] return 0.5f*exp(|x|) */
-	if (ix < 0x42b17217)
-		return 0.5f*expf(x);
-
-	/* |x| in [log(maxfloat), overflowthresold] */
-	if (ix <= 0x42b2d4fc)
-		return __expo2f(x);
-
-	/* |x| > overflowthresold or nan */
-	x *= 0x1p127f;
-	return x;
+	/* |x| > log(FLT_MAX) or nan */
+	t = __expo2f(x);
+	return t;
 }

src/math/coshl.c

-/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_coshl.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.
- * ====================================================
- */
-/* coshl(x)
- * Method :
- * mathematically coshl(x) if defined to be (exp(x)+exp(-x))/2
- *      1. Replace x by |x| (coshl(x) = coshl(-x)).
- *      2.
- *                                                      [ exp(x) - 1 ]^2
- *          0        <= x <= ln2/2  :  coshl(x) := 1 + -------------------
- *                                                         2*exp(x)
- *
- *                                                 exp(x) +  1/exp(x)
- *          ln2/2    <= x <= 22     :  coshl(x) := -------------------
- *                                                         2
- *          22       <= x <= lnovft :  coshl(x) := expl(x)/2
- *          lnovft   <= x <= ln2ovft:  coshl(x) := expl(x/2)/2 * expl(x/2)
- *          ln2ovft  <  x           :  coshl(x) := inf (overflow)
- *
- * Special cases:
- *      coshl(x) is |x| if x is +INF, -INF, or NaN.
- *      only coshl(0)=1 is exact for finite x.
- */
-
 #include "libm.h"
 
 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
@@ -45,41 +13,32 @@ long double coshl(long double x)
 		struct{uint64_t m; uint16_t se; uint16_t pad;} i;
 	} u = {.f = x};
 	unsigned ex = u.i.se & 0x7fff;
+	uint32_t w;
 	long double t;
-	uint32_t mx,lx;
 
 	/* |x| */
 	u.i.se = ex;
 	x = u.f;
-	mx = u.i.m >> 32;
-	lx = u.i.m;
+	w = u.i.m >> 32;
 
-	/* |x| in [0,0.5*ln2], return 1+expm1l(|x|)^2/(2*expl(|x|)) */
-	if (ex < 0x3fff-2 || (ex == 0x3fff-2 && mx < 0xb17217f7)) {
-		t = expm1l(x);
-		if (ex < 0x3fff-64)
+	/* |x| < log(2) */
+	if (ex < 0x3fff-1 || (ex == 0x3fff-1 && w < 0xb17217f7)) {
+		if (ex < 0x3fff-32) {
+			FORCE_EVAL(x + 0x1p120f);
 			return 1;
+		}
+		t = expm1l(x);
 		return 1 + t*t/(2*(1+t));
 	}
 
-	/* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|)/2; */
-	if (ex < 0x3fff+4 || (ex == 0x3fff+4 && mx < 0xb0000000)) {
+	/* |x| < log(LDBL_MAX) */
+	if (ex < 0x3fff+13 || (ex == 0x3fff+13 && w < 0xb17217f7)) {
 		t = expl(x);
-		return 0.5*t + 0.5/t;
-	}
-
-	/* |x| in [22, ln(maxdouble)] return 0.5*exp(|x|) */
-	if (ex < 0x3fff+13 || (ex == 0x3fff+13 && mx < 0xb1700000))
-		return 0.5*expl(x);
-
-	/* |x| in [log(maxdouble), log(2*maxdouble)) */
-	if (ex == 0x3fff+13 && (mx < 0xb174ddc0 ||
-	     (mx == 0xb174ddc0 && lx < 0x31aec0eb))) {
-		t = expl(0.5*x);
-		return 0.5*t*t;
+		return 0.5*(t + 1/t);
 	}
 
-	/* |x| >= log(2*maxdouble) or nan */
-	return x*0x1p16383L;
+	/* |x| > log(LDBL_MAX) or nan */
+	t = expl(0.5*x);
+	return 0.5*t*t;
 }
 #endif