vendor/libm/src/math/k_tan.rs

   1 // origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
   2 //
   3 // ====================================================
   4 // Copyright 2004 Sun Microsystems, Inc.  All Rights Reserved.
   5 //
   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 function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
  12 // Input x is assumed to be bounded by ~pi/4 in magnitude.
  13 // Input y is the tail of x.
  14 // Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
  15 //
  16 // Algorithm
  17 //      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  18 //      2. Callers must return tan(-0) = -0 without calling here since our
  19 //         odd polynomial is not evaluated in a way that preserves -0.
  20 //         Callers may do the optimization tan(x) ~ x for tiny x.
  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 static T: [f64; 13] = [
  44     3.33333333333334091986e-01,  /* 3FD55555, 55555563 */
  45     1.33333333333201242699e-01,  /* 3FC11111, 1110FE7A */
  46     5.39682539762260521377e-02,  /* 3FABA1BA, 1BB341FE */
  47     2.18694882948595424599e-02,  /* 3F9664F4, 8406D637 */
  48     8.86323982359930005737e-03,  /* 3F8226E3, E96E8493 */
  49     3.59207910759131235356e-03,  /* 3F6D6D22, C9560328 */
  50     1.45620945432529025516e-03,  /* 3F57DBC8, FEE08315 */
  51     5.88041240820264096874e-04,  /* 3F4344D8, F2F26501 */
  52     2.46463134818469906812e-04,  /* 3F3026F7, 1A8D1068 */
  53     7.81794442939557092300e-05,  /* 3F147E88, A03792A6 */
  54     7.14072491382608190305e-05,  /* 3F12B80F, 32F0A7E9 */
  55     -1.85586374855275456654e-05, /* BEF375CB, DB605373 */
  56     2.59073051863633712884e-05,  /* 3EFB2A70, 74BF7AD4 */
  57 ];
  58 const PIO4: f64 = 7.85398163397448278999e-01; /* 3FE921FB, 54442D18 */
  59 const PIO4_LO: f64 = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
  60
  61 #[inline]
  62 #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
  63 pub(crate) fn k_tan(mut x: f64, mut y: f64, odd: i32) -> f64 {
  64     let hx = (f64::to_bits(x) >> 32) as u32;
  65     let big = (hx & 0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
  66     if big {
  67         let sign = hx >> 31;
  68         if sign != 0 {
  69             x = -x;
  70             y = -y;
  71         }
  72         x = (PIO4 - x) + (PIO4_LO - y);
  73         y = 0.0;
  74     }
  75     let z = x * x;
  76     let w = z * z;
  77     /*
  78      * Break x^5*(T[1]+x^2*T[2]+...) into
  79      * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
  80      * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
  81      */
  82     let r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11]))));
  83     let v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12])))));
  84     let s = z * x;
  85     let r = y + z * (s * (r + v) + y) + s * T[0];
  86     let w = x + r;
  87     if big {
  88         let sign = hx >> 31;
  89         let s = 1.0 - 2.0 * odd as f64;
  90         let v = s - 2.0 * (x + (r - w * w / (w + s)));
  91         return if sign != 0 { -v } else { v };
  92     }
  93     if odd == 0 {
  94         return w;
  95     }
  96     /* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
  97     let w0 = zero_low_word(w);
  98     let v = r - (w0 - x); /* w0+v = r+x */
  99     let a = -1.0 / w;
 100     let a0 = zero_low_word(a);
 101     a0 + a * (1.0 + a0 * w0 + a0 * v)
 102 }
 103
 104 #[inline]
 105 fn zero_low_word(x: f64) -> f64 {
 106     f64::from_bits(f64::to_bits(x) & 0xFFFF_FFFF_0000_0000)
 107 }