k_cos.c 2.8 KB

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182
  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. * __kernel_cos( x, y )
  13. * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
  14. * Input x is assumed to be bounded by ~pi/4 in magnitude.
  15. * Input y is the tail of x.
  16. *
  17. * Algorithm
  18. * 1. Since cos(-x) = cos(x), we need only to consider positive x.
  19. * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
  20. * 3. cos(x) is approximated by a polynomial of degree 14 on
  21. * [0,pi/4]
  22. * 4 14
  23. * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
  24. * where the remez error is
  25. *
  26. * | 2 4 6 8 10 12 14 | -58
  27. * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
  28. * | |
  29. *
  30. * 4 6 8 10 12 14
  31. * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
  32. * cos(x) = 1 - x*x/2 + r
  33. * since cos(x+y) ~ cos(x) - sin(x)*y
  34. * ~ cos(x) - x*y,
  35. * a correction term is necessary in cos(x) and hence
  36. * cos(x+y) = 1 - (x*x/2 - (r - x*y))
  37. * For better accuracy when x > 0.3, let qx = |x|/4 with
  38. * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
  39. * Then
  40. * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
  41. * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
  42. * magnitude of the latter is at least a quarter of x*x/2,
  43. * thus, reducing the rounding error in the subtraction.
  44. */
  45. #include "math.h"
  46. #include "math_private.h"
  47. static const double
  48. one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  49. C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
  50. C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
  51. C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
  52. C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
  53. C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
  54. C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
  55. double attribute_hidden __kernel_cos(double x, double y)
  56. {
  57. double a,hz,z,r,qx;
  58. int32_t ix;
  59. GET_HIGH_WORD(ix,x);
  60. ix &= 0x7fffffff; /* ix = |x|'s high word*/
  61. if(ix<0x3e400000) { /* if x < 2**27 */
  62. if(((int)x)==0) return one; /* generate inexact */
  63. }
  64. z = x*x;
  65. r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
  66. if(ix < 0x3FD33333) /* if |x| < 0.3 */
  67. return one - (0.5*z - (z*r - x*y));
  68. else {
  69. if(ix > 0x3fe90000) { /* x > 0.78125 */
  70. qx = 0.28125;
  71. } else {
  72. INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */
  73. }
  74. hz = 0.5*z-qx;
  75. a = one-qx;
  76. return a - (hz - (z*r-x*y));
  77. }
  78. }