1 /* $OpenBSD: s_casinl.c,v 1.5 2016/09/12 19:47:02 guenther Exp $ */ 2 3 /* 4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 5 * 6 * Permission to use, copy, modify, and distribute this software for any 7 * purpose with or without fee is hereby granted, provided that the above 8 * copyright notice and this permission notice appear in all copies. 9 * 10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 17 */ 18 19 /* casinl() 20 * 21 * Complex circular arc sine 22 * 23 * 24 * 25 * SYNOPSIS: 26 * 27 * long double complex casinl(); 28 * long double complex z, w; 29 * 30 * w = casinl( z ); 31 * 32 * 33 * 34 * DESCRIPTION: 35 * 36 * Inverse complex sine: 37 * 38 * 2 39 * w = -i clog( iz + csqrt( 1 - z ) ). 40 * 41 * 42 * ACCURACY: 43 * 44 * Relative error: 45 * arithmetic domain # trials peak rms 46 * DEC -10,+10 10100 2.1e-15 3.4e-16 47 * IEEE -10,+10 30000 2.2e-14 2.7e-15 48 * Larger relative error can be observed for z near zero. 49 * Also tested by csin(casin(z)) = z. 50 */ 51 52 #include <complex.h> 53 #include <float.h> 54 #include <math.h> 55 56 #if LDBL_MANT_DIG == 64 57 static const long double MACHEPL= 5.42101086242752217003726400434970855712890625E-20L; 58 #elif LDBL_MANT_DIG == 113 59 static const long double MACHEPL = 9.629649721936179265279889712924636592690508e-35L; 60 #endif 61 62 static const long double PIO2L = 1.570796326794896619231321691639751442098585L; 63 64 long double complex 65 casinl(long double complex z) 66 { 67 long double complex w; 68 long double x, y, b; 69 static long double complex ca, ct, zz, z2; 70 71 x = creall(z); 72 y = cimagl(z); 73 74 #if 0 75 if (y == 0.0L) { 76 if (fabsl(x) > 1.0L) { 77 w = PIO2L + 0.0L * I; 78 /*mtherr( "casinl", DOMAIN );*/ 79 } 80 else { 81 w = asinl(x) + 0.0L * I; 82 } 83 return (w); 84 } 85 #endif 86 87 /* Power series expansion */ 88 b = cabsl(z); 89 if (b < 0.125L) { 90 long double complex sum; 91 long double n, cn; 92 93 z2 = (x - y) * (x + y) + (2.0L * x * y) * I; 94 cn = 1.0L; 95 n = 1.0L; 96 ca = x + y * I; 97 sum = x + y * I; 98 do { 99 ct = z2 * ca; 100 ca = ct; 101 102 cn *= n; 103 n += 1.0L; 104 cn /= n; 105 n += 1.0L; 106 b = cn/n; 107 108 ct *= b; 109 sum += ct; 110 b = cabsl(ct); 111 } 112 113 while (b > MACHEPL); 114 w = sum; 115 return w; 116 } 117 118 ca = x + y * I; 119 ct = ca * I; /* iz */ 120 /* sqrt(1 - z*z) */ 121 /* cmul(&ca, &ca, &zz) */ 122 /* x * x - y * y */ 123 zz = (x - y) * (x + y) + (2.0L * x * y) * I; 124 zz = 1.0L - creall(zz) - cimagl(zz) * I; 125 z2 = csqrtl(zz); 126 127 zz = ct + z2; 128 zz = clogl(zz); 129 /* multiply by 1/i = -i */ 130 w = zz * (-1.0L * I); 131 return (w); 132 } 133 DEF_STD(casinl); 134