k_tan.c 3.9 KB

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  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. /* __kernel_tan( x, y, k )
  12. * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  13. * Input x is assumed to be bounded by ~pi/4 in magnitude.
  14. * Input y is the tail of x.
  15. * Input k indicates whether tan (if k=1) or
  16. * -1/tan (if k= -1) is returned.
  17. *
  18. * Algorithm
  19. * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
  20. * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
  21. * 3. tan(x) is approximated by a odd polynomial of degree 27 on
  22. * [0,0.67434]
  23. * 3 27
  24. * tan(x) ~ x + T1*x + ... + T13*x
  25. * where
  26. *
  27. * |tan(x) 2 4 26 | -59.2
  28. * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
  29. * | x |
  30. *
  31. * Note: tan(x+y) = tan(x) + tan'(x)*y
  32. * ~ tan(x) + (1+x*x)*y
  33. * Therefore, for better accuracy in computing tan(x+y), let
  34. * 3 2 2 2 2
  35. * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
  36. * then
  37. * 3 2
  38. * tan(x+y) = x + (T1*x + (x *(r+y)+y))
  39. *
  40. * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
  41. * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  42. * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  43. */
  44. #include "math.h"
  45. #include "math_private.h"
  46. static const double
  47. one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  48. pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
  49. pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
  50. T[] = {
  51. 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
  52. 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
  53. 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
  54. 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
  55. 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
  56. 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
  57. 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
  58. 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
  59. 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
  60. 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
  61. 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
  62. -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
  63. 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
  64. };
  65. double attribute_hidden __kernel_tan(double x, double y, int iy)
  66. {
  67. double z,r,v,w,s;
  68. int32_t ix,hx;
  69. GET_HIGH_WORD(hx,x);
  70. ix = hx&0x7fffffff; /* high word of |x| */
  71. if(ix<0x3e300000) /* x < 2**-28 */
  72. {if((int)x==0) { /* generate inexact */
  73. u_int32_t low;
  74. GET_LOW_WORD(low,x);
  75. if(((ix|low)|(iy+1))==0) return one/fabs(x);
  76. else return (iy==1)? x: -one/x;
  77. }
  78. }
  79. if(ix>=0x3FE59428) { /* |x|>=0.6744 */
  80. if(hx<0) {x = -x; y = -y;}
  81. z = pio4-x;
  82. w = pio4lo-y;
  83. x = z+w; y = 0.0;
  84. }
  85. z = x*x;
  86. w = z*z;
  87. /* Break x^5*(T[1]+x^2*T[2]+...) into
  88. * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
  89. * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
  90. */
  91. r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
  92. v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
  93. s = z*x;
  94. r = y + z*(s*(r+v)+y);
  95. r += T[0]*s;
  96. w = x+r;
  97. if(ix>=0x3FE59428) {
  98. v = (double)iy;
  99. return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
  100. }
  101. if(iy==1) return w;
  102. else { /* if allow error up to 2 ulp,
  103. simply return -1.0/(x+r) here */
  104. /* compute -1.0/(x+r) accurately */
  105. double a,t;
  106. z = w;
  107. SET_LOW_WORD(z,0);
  108. v = r-(z - x); /* z+v = r+x */
  109. t = a = -1.0/w; /* a = -1.0/w */
  110. SET_LOW_WORD(t,0);
  111. s = 1.0+t*z;
  112. return t+a*(s+t*v);
  113. }
  114. }