-/* 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) := huge*huge (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"
-static const double one = 1.0, half = 0.5, huge = 1.0e300;
-
+/* 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)
{
- double t, w;
- int32_t ix;
-
- GET_HIGH_WORD(ix, x);
- ix &= 0x7fffffff;
-
- /* x is INF or NaN */
- if (ix >= 0x7ff00000)
- return x*x;
-
- /* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
- if (ix < 0x3fd62e43) {
- t = expm1(fabs(x));
- w = one+t;
- if (ix < 0x3c800000)
- return w; /* cosh(tiny) = 1 */
- return one + (t*t)/(w+w);
+ union {double f; uint64_t i;} u = {.f = x};
+ uint32_t w;
+ double t;
+
+ /* |x| */
+ u.i &= (uint64_t)-1/2;
+ x = u.f;
+ w = u.i >> 32;
+
+ /* |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) {
- t = exp(fabs(x));
- return half*t + half/t;
+ /* |x| < log(DBL_MAX) */
+ if (w < 0x40862e42) {
+ t = exp(x);
+ /* note: if x>log(0x1p26) then the 1/t is not needed */
+ return 0.5*(t + 1/t);
}
- /* |x| in [22, log(maxdouble)] return half*exp(|x|) */
- if (ix < 0x40862E42)
- return half*exp(fabs(x));
-
- /* |x| in [log(maxdouble), overflowthresold] */
- if (ix <= 0x408633CE)
- return __expo2(fabs(x));
-
- /* |x| > overflowthresold, cosh(x) overflow */
- return huge*huge;
+ /* |x| > log(DBL_MAX) or nan */
+ /* note: the result is stored to handle overflow */
+ t = __expo2(x, 1.0);
+ return t;
}