1#include "libm.h"
2
3/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
4double asinh(double x) {
5    union {
6        double f;
7        uint64_t i;
8    } u = {.f = x};
9    unsigned e = u.i >> 52 & 0x7ff;
10    unsigned s = u.i >> 63;
11
12    /* |x| */
13    u.i &= (uint64_t)-1 / 2;
14    x = u.f;
15
16    if (e >= 0x3ff + 26) {
17        /* |x| >= 0x1p26 or inf or nan */
18        x = log(x) + 0.693147180559945309417232121458176568;
19    } else if (e >= 0x3ff + 1) {
20        /* |x| >= 2 */
21        x = log(2 * x + 1 / (sqrt(x * x + 1) + x));
22    } else if (e >= 0x3ff - 26) {
23        /* |x| >= 0x1p-26, up to 1.6ulp error in [0.125,0.5] */
24        x = log1p(x + x * x / (sqrt(x * x + 1) + 1));
25    } else {
26        /* |x| < 0x1p-26, raise inexact if x != 0 */
27        FORCE_EVAL(x + 0x1p120f);
28    }
29    return s ? -x : x;
30}
31