1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* atan(x) 13 * Method 14 * 1. Reduce x to positive by atan(x) = -atan(-x). 15 * 2. According to the integer k=4t+0.25 chopped, t=x, the argument 16 * is further reduced to one of the following intervals and the 17 * arctangent of t is evaluated by the corresponding formula: 18 * 19 * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) 20 * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) ) 21 * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) ) 22 * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) ) 23 * [39/16,INF] atan(x) = atan(INF) + atan( -1/t ) 24 * 25 * Constants: 26 * The hexadecimal values are the intended ones for the following 27 * constants. The decimal values may be used, provided that the 28 * compiler will convert from decimal to binary accurately enough 29 * to produce the hexadecimal values shown. 30 */ 31 32 #include <float.h> 33 34 #include "math.h" 35 #include "math_private.h" 36 37 static const double atanhi[] = { 38 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */ 39 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */ 40 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */ 41 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */ 42 }; 43 44 static const double atanlo[] = { 45 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */ 46 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */ 47 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */ 48 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */ 49 }; 50 51 static const double aT[] = { 52 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */ 53 -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */ 54 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */ 55 -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */ 56 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */ 57 -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */ 58 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */ 59 -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */ 60 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */ 61 -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */ 62 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */ 63 }; 64 65 static const double 66 one = 1.0, 67 huge = 1.0e300; 68 69 double 70 atan(double x) 71 { 72 double w,s1,s2,z; 73 int32_t ix,hx,id; 74 75 GET_HIGH_WORD(hx,x); 76 ix = hx&0x7fffffff; 77 if(ix>=0x44100000) { /* if |x| >= 2^66 */ 78 u_int32_t low; 79 GET_LOW_WORD(low,x); 80 if(ix>0x7ff00000|| 81 (ix==0x7ff00000&&(low!=0))) 82 return x+x; /* NaN */ 83 if(hx>0) return atanhi[3]+*(volatile double *)&atanlo[3]; 84 else return -atanhi[3]-*(volatile double *)&atanlo[3]; 85 } if (ix < 0x3fdc0000) { /* |x| < 0.4375 */ 86 if (ix < 0x3e400000) { /* |x| < 2^-27 */ 87 if(huge+x>one) return x; /* raise inexact */ 88 } 89 id = -1; 90 } else { 91 x = fabs(x); 92 if (ix < 0x3ff30000) { /* |x| < 1.1875 */ 93 if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */ 94 id = 0; x = (2.0*x-one)/(2.0+x); 95 } else { /* 11/16<=|x|< 19/16 */ 96 id = 1; x = (x-one)/(x+one); 97 } 98 } else { 99 if (ix < 0x40038000) { /* |x| < 2.4375 */ 100 id = 2; x = (x-1.5)/(one+1.5*x); 101 } else { /* 2.4375 <= |x| < 2^66 */ 102 id = 3; x = -1.0/x; 103 } 104 }} 105 /* end of argument reduction */ 106 z = x*x; 107 w = z*z; 108 /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */ 109 s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10]))))); 110 s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9])))); 111 if (id<0) return x - x*(s1+s2); 112 else { 113 z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x); 114 return (hx<0)? -z:z; 115 } 116 } 117 118 #if LDBL_MANT_DIG == 53 119 __weak_reference(atan, atanl); 120 #endif 121