sqrt.rs - source
Description: Source of the Rust file `/Users/erlendbasso/.cargo/registry/src/github.com-1ecc6299db9ec823/libm-0.2.6/src/math/sqrt.rs`.
Keywords: rust, rustlang, rust-lang

    1
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
    50
    51
    52
    53
    54
    55
    56
    57
    58
    59
    60
    61
    62
    63
    64
    65
    66
    67
    68
    69
    70
    71
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    82
    83
    84
    85
    86
    87
    88
    89
    90
    91
    92
    93
    94
    95
    96
    97
    98
    99
    100
    101
    102
    103
    104
    105
    106
    107
    108
    109
    110
    111
    112
    113
    114
    115
    116
    117
    118
    119
    120
    121
    122
    123
    124
    125
    126
    127
    128
    129
    130
    131
    132
    133
    134
    135
    136
    137
    138
    139
    140
    141
    142
    143
    144
    145
    146
    147
    148
    149
    150
    151
    152
    153
    154
    155
    156
    157
    158
    159
    160
    161
    162
    163
    164
    165
    166
    167
    168
    169
    170
    171
    172
    173
    174
    175
    176
    177
    178
    179
    180
    181
    182
    183
    184
    185
    186
    187
    188
    189
    190
    191
    192
    193
    194
    195
    196
    197
    198
    199
    200
    201
    202
    203
    204
    205
    206
    207
    208
    209
    210
    211
    212
    213
    214
    215
    216
    217
    218
    219
    220
    221
    222
    223
    224
    225
    226
    227
    228
    229
    230
    231
    232
    233
    234
    235
    236
    237
    238
    239
    240
    241
    242
    243
    244
    245
    246
    247
    248
    249
    250
    251
    252
    253
    254
    255
    256
    257
    258
    259
    260
    261
    262
    263
    264

    /* origin: FreeBSD /usr/src/lib/msun/src/e_sqrt.c */
    /*
     * ====================================================
     * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     *
     * Developed at SunSoft, a Sun Microsystems, Inc. business.
     * Permission to use, copy, modify, and distribute this
     * software is freely granted, provided that this notice
     * is preserved.
     * ====================================================
     */
    /* sqrt(x)
     * Return correctly rounded sqrt.
     *           ------------------------------------------
     *           |  Use the hardware sqrt if you have one |
     *           ------------------------------------------
     * Method:
     *   Bit by bit method using integer arithmetic. (Slow, but portable)
     *   1. Normalization
     *      Scale x to y in [1,4) with even powers of 2:
     *      find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
     *              sqrt(x) = 2^k * sqrt(y)
     *   2. Bit by bit computation
     *      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
     *           i                                                   0
     *                                     i+1         2
     *          s  = 2*q , and      y  =  2   * ( y - q  ).         (1)
     *           i      i            i                 i
     *
     *      To compute q    from q , one checks whether
     *                  i+1       i
     *
     *                            -(i+1) 2
     *                      (q + 2      ) <= y.                     (2)
     *                        i
     *                                                            -(i+1)
     *      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
     *                             i+1   i             i+1   i
     *
     *      With some algebraic manipulation, it is not difficult to see
     *      that (2) is equivalent to
     *                             -(i+1)
     *                      s  +  2       <= y                      (3)
     *                       i                i
     *
     *      The advantage of (3) is that s  and y  can be computed by
     *                                    i      i
     *      the following recurrence formula:
     *          if (3) is false
     *
     *          s     =  s  ,       y    = y   ;                    (4)
     *           i+1      i          i+1    i
     *
     *          otherwise,
     *                         -i                     -(i+1)
     *          s     =  s  + 2  ,  y    = y  -  s  - 2             (5)
     *           i+1      i          i+1    i     i
     *
     *      One may easily use induction to prove (4) and (5).
     *      Note. Since the left hand side of (3) contain only i+2 bits,
     *            it does not necessary to do a full (53-bit) comparison
     *            in (3).
     *   3. Final rounding
     *      After generating the 53 bits result, we compute one more bit.
     *      Together with the remainder, we can decide whether the
     *      result is exact, bigger than 1/2ulp, or less than 1/2ulp
     *      (it will never equal to 1/2ulp).
     *      The rounding mode can be detected by checking whether
     *      huge + tiny is equal to huge, and whether huge - tiny is
     *      equal to huge for some floating point number "huge" and "tiny".
     *
     * Special cases:
     *      sqrt(+-0) = +-0         ... exact
     *      sqrt(inf) = inf
     *      sqrt(-ve) = NaN         ... with invalid signal
     *      sqrt(NaN) = NaN         ... with invalid signal for signaling NaN
     */

    use core::f64;

    #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
    pub fn sqrt(x: f64) -> f64 {
        // On wasm32 we know that LLVM's intrinsic will compile to an optimized
        // `f64.sqrt` native instruction, so we can leverage this for both code size
        // and speed.
        llvm_intrinsically_optimized! {
            #[cfg(target_arch = "wasm32")] {
                return if x < 0.0 {
                    f64::NAN
                } else {
                    unsafe { ::core::intrinsics::sqrtf64(x) }
                }
            }
        }
        #[cfg(target_feature = "sse2")]
        {
            // Note: This path is unlikely since LLVM will usually have already
            // optimized sqrt calls into hardware instructions if sse2 is available,
            // but if someone does end up here they'll apprected the speed increase.
            #[cfg(target_arch = "x86")]
            use core::arch::x86::*;
            #[cfg(target_arch = "x86_64")]
            use core::arch::x86_64::*;
            unsafe {
                let m = _mm_set_sd(x);
                let m_sqrt = _mm_sqrt_pd(m);
                _mm_cvtsd_f64(m_sqrt)
            }
        }
        #[cfg(not(target_feature = "sse2"))]
        {
            use core::num::Wrapping;

            const TINY: f64 = 1.0e-300;

            let mut z: f64;
            let sign: Wrapping<u32> = Wrapping(0x80000000);
            let mut ix0: i32;
            let mut s0: i32;
            let mut q: i32;
            let mut m: i32;
            let mut t: i32;
            let mut i: i32;
            let mut r: Wrapping<u32>;
            let mut t1: Wrapping<u32>;
            let mut s1: Wrapping<u32>;
            let mut ix1: Wrapping<u32>;
            let mut q1: Wrapping<u32>;

            ix0 = (x.to_bits() >> 32) as i32;
            ix1 = Wrapping(x.to_bits() as u32);

            /* take care of Inf and NaN */
            if (ix0 & 0x7ff00000) == 0x7ff00000 {
                return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */
            }
            /* take care of zero */
            if ix0 <= 0 {
                if ((ix0 & !(sign.0 as i32)) | ix1.0 as i32) == 0 {
                    return x; /* sqrt(+-0) = +-0 */
                }
                if ix0 < 0 {
                    return (x - x) / (x - x); /* sqrt(-ve) = sNaN */
                }
            }
            /* normalize x */
            m = ix0 >> 20;
            if m == 0 {
                /* subnormal x */
                while ix0 == 0 {
                    m -= 21;
                    ix0 |= (ix1 >> 11).0 as i32;
                    ix1 <<= 21;
                }
                i = 0;
                while (ix0 & 0x00100000) == 0 {
                    i += 1;
                    ix0 <<= 1;
                }
                m -= i - 1;
                ix0 |= (ix1 >> (32 - i) as usize).0 as i32;
                ix1 = ix1 << i as usize;
            }
            m -= 1023; /* unbias exponent */
            ix0 = (ix0 & 0x000fffff) | 0x00100000;
            if (m & 1) == 1 {
                /* odd m, double x to make it even */
                ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32;
                ix1 += ix1;
            }
            m >>= 1; /* m = [m/2] */

            /* generate sqrt(x) bit by bit */
            ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32;
            ix1 += ix1;
            q = 0; /* [q,q1] = sqrt(x) */
            q1 = Wrapping(0);
            s0 = 0;
            s1 = Wrapping(0);
            r = Wrapping(0x00200000); /* r = moving bit from right to left */

            while r != Wrapping(0) {
                t = s0 + r.0 as i32;
                if t <= ix0 {
                    s0 = t + r.0 as i32;
                    ix0 -= t;
                    q += r.0 as i32;
                }
                ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32;
                ix1 += ix1;
                r >>= 1;
            }

            r = sign;
            while r != Wrapping(0) {
                t1 = s1 + r;
                t = s0;
                if t < ix0 || (t == ix0 && t1 <= ix1) {
                    s1 = t1 + r;
                    if (t1 & sign) == sign && (s1 & sign) == Wrapping(0) {
                        s0 += 1;
                    }
                    ix0 -= t;
                    if ix1 < t1 {
                        ix0 -= 1;
                    }
                    ix1 -= t1;
                    q1 += r;
                }
                ix0 += ix0 + ((ix1 & sign) >> 31).0 as i32;
                ix1 += ix1;
                r >>= 1;
            }

            /* use floating add to find out rounding direction */
            if (ix0 as u32 | ix1.0) != 0 {
                z = 1.0 - TINY; /* raise inexact flag */
                if z >= 1.0 {
                    z = 1.0 + TINY;
                    if q1.0 == 0xffffffff {
                        q1 = Wrapping(0);
                        q += 1;
                    } else if z > 1.0 {
                        if q1.0 == 0xfffffffe {
                            q += 1;
                        }
                        q1 += Wrapping(2);
                    } else {
                        q1 += q1 & Wrapping(1);
                    }
                }
            }
            ix0 = (q >> 1) + 0x3fe00000;
            ix1 = q1 >> 1;
            if (q & 1) == 1 {
                ix1 |= sign;
            }
            ix0 += m << 20;
            f64::from_bits((ix0 as u64) << 32 | ix1.0 as u64)
        }
    }

    #[cfg(test)]
    mod tests {
        use super::*;
        use core::f64::*;

        #[test]
        fn sanity_check() {
            assert_eq!(sqrt(100.0), 10.0);
            assert_eq!(sqrt(4.0), 2.0);
        }

        /// The spec: https://en.cppreference.com/w/cpp/numeric/math/sqrt
        #[test]
        fn spec_tests() {
            // Not Asserted: FE_INVALID exception is raised if argument is negative.
            assert!(sqrt(-1.0).is_nan());
            assert!(sqrt(NAN).is_nan());
            for f in [0.0, -0.0, INFINITY].iter().copied() {
                assert_eq!(sqrt(f), f);
            }
        }
    }