math: rewrite inverse hyperbolic functions to be simpler/smaller

modifications:
* avoid unsigned->signed integer conversion
* do not handle special cases when they work correctly anyway
* more strict threshold values (0x1p26 instead of 0x1p28 etc)
* smaller code, cleaner branching logic
* same precision as the old code:
    acosh(x) has up to 2ulp error in [1,1.125]
    asinh(x) has up to 1.6ulp error in [0.125,0.5], [-0.5,-0.125]
    atanh(x) has up to 1.7ulp error in [0.125,0.5], [-0.5,-0.125]

1 files changed, 13 insertions, 48 deletions

diff --git a/src/math/acosh.c b/src/math/acosh.c
index 15f51c6e..4ce9b3d1 100644
--- a/src/math/acosh.c
+++ b/src/math/acosh.c
@@ -1,54 +1,19 @@
-/* origin: FreeBSD /usr/src/lib/msun/src/e_acosh.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.
- * ====================================================
- *
- */
-/* acosh(x)
- * Method :
- *      Based on
- *              acosh(x) = log [ x + sqrt(x*x-1) ]
- *      we have
- *              acosh(x) := log(x)+ln2, if x is large; else
- *              acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
- *              acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
- *
- * Special cases:
- *      acosh(x) is NaN with signal if x<1.
- *      acosh(NaN) is NaN without signal.
- */
-
 #include "libm.h"
 
-static const double
-ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */
-
+/* acosh(x) = log(x + sqrt(x*x-1)) */
 double acosh(double x)
 {
-	double t;
-	int32_t hx;
-	uint32_t lx;
+	union {double f; uint64_t i;} u = {.f = x};
+	unsigned e = u.i >> 52 & 0x7ff;
+
+	/* x < 1 domain error is handled in the called functions */
 
-	EXTRACT_WORDS(hx, lx, x);
-	if (hx < 0x3ff00000) {  /* x < 1 */
-		return (x-x)/(x-x);
-	} else if (hx >= 0x41b00000) {  /* x > 2**28 */
-		if (hx >= 0x7ff00000)  /* x is inf of NaN */
-			return x+x;
-		return log(x) + ln2;   /* acosh(huge) = log(2x) */
-	} else if ((hx-0x3ff00000 | lx) == 0) {
-		return 0.0;            /* acosh(1) = 0 */
-	} else if (hx > 0x40000000) {  /* 2**28 > x > 2 */
-		t = x*x;
-		return log(2.0*x - 1.0/(x+sqrt(t-1.0)));
-	} else {                /* 1 < x < 2 */
-		t = x-1.0;
-		return log1p(t + sqrt(2.0*t+t*t));
-	}
+	if (e < 0x3ff + 1)
+		/* |x| < 2, up to 2ulp error in [1,1.125] */
+		return log1p(x-1 + sqrt((x-1)*(x-1)+2*(x-1)));
+	if (e < 0x3ff + 26)
+		/* |x| < 0x1p26 */
+		return log(2*x - 1/(x+sqrt(x*x-1)));
+	/* |x| >= 0x1p26 or nan */
+	return log(x) + 0.693147180559945309417232121458176568;
 }