e_log.c 4.2 KB

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129
  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. /* __ieee754_log(x)
  12. * Return the logrithm of x
  13. *
  14. * Method :
  15. * 1. Argument Reduction: find k and f such that
  16. * x = 2^k * (1+f),
  17. * where sqrt(2)/2 < 1+f < sqrt(2) .
  18. *
  19. * 2. Approximation of log(1+f).
  20. * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  21. * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  22. * = 2s + s*R
  23. * We use a special Reme algorithm on [0,0.1716] to generate
  24. * a polynomial of degree 14 to approximate R The maximum error
  25. * of this polynomial approximation is bounded by 2**-58.45. In
  26. * other words,
  27. * 2 4 6 8 10 12 14
  28. * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
  29. * (the values of Lg1 to Lg7 are listed in the program)
  30. * and
  31. * | 2 14 | -58.45
  32. * | Lg1*s +...+Lg7*s - R(z) | <= 2
  33. * | |
  34. * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  35. * In order to guarantee error in log below 1ulp, we compute log
  36. * by
  37. * log(1+f) = f - s*(f - R) (if f is not too large)
  38. * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
  39. *
  40. * 3. Finally, log(x) = k*ln2 + log(1+f).
  41. * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  42. * Here ln2 is split into two floating point number:
  43. * ln2_hi + ln2_lo,
  44. * where n*ln2_hi is always exact for |n| < 2000.
  45. *
  46. * Special cases:
  47. * log(x) is NaN with signal if x < 0 (including -INF) ;
  48. * log(+INF) is +INF; log(0) is -INF with signal;
  49. * log(NaN) is that NaN with no signal.
  50. *
  51. * Accuracy:
  52. * according to an error analysis, the error is always less than
  53. * 1 ulp (unit in the last place).
  54. *
  55. * Constants:
  56. * The hexadecimal values are the intended ones for the following
  57. * constants. The decimal values may be used, provided that the
  58. * compiler will convert from decimal to binary accurately enough
  59. * to produce the hexadecimal values shown.
  60. */
  61. #include "math.h"
  62. #include "math_private.h"
  63. static const double
  64. ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
  65. ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
  66. two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
  67. Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
  68. Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
  69. Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
  70. Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
  71. Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
  72. Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
  73. Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
  74. static const double zero = 0.0;
  75. double __ieee754_log(double x)
  76. {
  77. double hfsq,f,s,z,R,w,t1,t2,dk;
  78. int32_t k,hx,i,j;
  79. u_int32_t lx;
  80. EXTRACT_WORDS(hx,lx,x);
  81. k=0;
  82. if (hx < 0x00100000) { /* x < 2**-1022 */
  83. if (((hx&0x7fffffff)|lx)==0)
  84. return -two54/zero; /* log(+-0)=-inf */
  85. if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
  86. k -= 54; x *= two54; /* subnormal number, scale up x */
  87. GET_HIGH_WORD(hx,x);
  88. }
  89. if (hx >= 0x7ff00000) return x+x;
  90. k += (hx>>20)-1023;
  91. hx &= 0x000fffff;
  92. i = (hx+0x95f64)&0x100000;
  93. SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
  94. k += (i>>20);
  95. f = x-1.0;
  96. if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
  97. if(f==zero) {if(k==0) return zero; else {dk=(double)k;
  98. return dk*ln2_hi+dk*ln2_lo;}
  99. }
  100. R = f*f*(0.5-0.33333333333333333*f);
  101. if(k==0) return f-R; else {dk=(double)k;
  102. return dk*ln2_hi-((R-dk*ln2_lo)-f);}
  103. }
  104. s = f/(2.0+f);
  105. dk = (double)k;
  106. z = s*s;
  107. i = hx-0x6147a;
  108. w = z*z;
  109. j = 0x6b851-hx;
  110. t1= w*(Lg2+w*(Lg4+w*Lg6));
  111. t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
  112. i |= j;
  113. R = t2+t1;
  114. if(i>0) {
  115. hfsq=0.5*f*f;
  116. if(k==0) return f-(hfsq-s*(hfsq+R)); else
  117. return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
  118. } else {
  119. if(k==0) return f-s*(f-R); else
  120. return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
  121. }
  122. }