sva-formula 0.14.31

Laws in t and f: the normal form, the Fourier dual rules, and the observations a law answers without sampling
Documentation 
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// Concern: the natural exponential every crate calls, the same IEEE operations on every target | Non-concern: where it underflows (underflow.rs) | IO: (f64) -> f64

//! Szabolcs Nagy's table method (ARM optimized-routines, also glibc's and musl's): with
//! `x = k ln2/128 + r`, `|r| <= ln2/256`, `e^x = 2^(k/128) (1 + tail) e^r`. Within 0.511 ulp
//! of `e^x` everywhere, FMA or not; libm's own `exp` holds 1 ulp and takes a division.

const N: u64 = 128;
const INV_LN2_N: f64 = f64::from_bits(0x40671547652b82fe);
/// 1.5 * 2^52: adding it rounds to an integer held in the low bits.
const SHIFT: f64 = f64::from_bits(0x4338000000000000);
/// ln2/128 in two parts, the high one short enough that `k` times it is exact.
const NEG_LN2_HI_N: f64 = -f64::from_bits(0x3f762e42fefa0000);
const NEG_LN2_LO_N: f64 = -f64::from_bits(0x3d0cf79abc9e3b3a);
/// `e^r - 1 - r` to within 1.09 * 2^-65 on `|r| <= ln2/256 + 2^-15`.
const C2: f64 = f64::from_bits(0x3fdffffffffffdbd);
const C3: f64 = f64::from_bits(0x3fc555555555543c);
const C4: f64 = f64::from_bits(0x3fa55555cf172b91);
const C5: f64 = f64::from_bits(0x3f81111167a4d017);

/// Per `i`: the bits of `tail` and of `2^(i/128)` rounded, less `i << 45`, so adding
/// `k << 45` lands `k / 128` in the exponent.
static T: [[u64; 2]; N as usize] = [
    [0x0000000000000000, 0x3ff0000000000000],
    [0x3c9b3b4f1a88bf6e, 0x3feff63da9fb3335],
    [0xbc7160139cd8dc5d, 0x3fefec9a3e778061],
    [0xbc905e7a108766d1, 0x3fefe315e86e7f85],
    [0x3c8cd2523567f613, 0x3fefd9b0d3158574],
    [0xbc8bce8023f98efa, 0x3fefd06b29ddf6de],
    [0x3c60f74e61e6c861, 0x3fefc74518759bc8],
    [0x3c90a3e45b33d399, 0x3fefbe3ecac6f383],
    [0x3c979aa65d837b6d, 0x3fefb5586cf9890f],
    [0x3c8eb51a92fdeffc, 0x3fefac922b7247f7],
    [0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2],
    [0xbc6a033489906e0b, 0x3fef9b66affed31b],
    [0xbc9556522a2fbd0e, 0x3fef9301d0125b51],
    [0xbc5080ef8c4eea55, 0x3fef8abdc06c31cc],
    [0xbc91c923b9d5f416, 0x3fef829aaea92de0],
    [0x3c80d3e3e95c55af, 0x3fef7a98c8a58e51],
    [0xbc801b15eaa59348, 0x3fef72b83c7d517b],
    [0xbc8f1ff055de323d, 0x3fef6af9388c8dea],
    [0x3c8b898c3f1353bf, 0x3fef635beb6fcb75],
    [0xbc96d99c7611eb26, 0x3fef5be084045cd4],
    [0x3c9aecf73e3a2f60, 0x3fef54873168b9aa],
    [0xbc8fe782cb86389d, 0x3fef4d5022fcd91d],
    [0x3c8a6f4144a6c38d, 0x3fef463b88628cd6],
    [0x3c807a05b0e4047d, 0x3fef3f49917ddc96],
    [0x3c968efde3a8a894, 0x3fef387a6e756238],
    [0x3c875e18f274487d, 0x3fef31ce4fb2a63f],
    [0x3c80472b981fe7f2, 0x3fef2b4565e27cdd],
    [0xbc96b87b3f71085e, 0x3fef24dfe1f56381],
    [0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1],
    [0xbc3d219b1a6fbffa, 0x3fef187fd0dad990],
    [0x3c8b3782720c0ab4, 0x3fef1285a6e4030b],
    [0x3c6e149289cecb8f, 0x3fef0cafa93e2f56],
    [0x3c834d754db0abb6, 0x3fef06fe0a31b715],
    [0x3c864201e2ac744c, 0x3fef0170fc4cd831],
    [0x3c8fdd395dd3f84a, 0x3feefc08b26416ff],
    [0xbc86a3803b8e5b04, 0x3feef6c55f929ff1],
    [0xbc924aedcc4b5068, 0x3feef1a7373aa9cb],
    [0xbc9907f81b512d8e, 0x3feeecae6d05d866],
    [0xbc71d1e83e9436d2, 0x3feee7db34e59ff7],
    [0xbc991919b3ce1b15, 0x3feee32dc313a8e5],
    [0x3c859f48a72a4c6d, 0x3feedea64c123422],
    [0xbc9312607a28698a, 0x3feeda4504ac801c],
    [0xbc58a78f4817895b, 0x3feed60a21f72e2a],
    [0xbc7c2c9b67499a1b, 0x3feed1f5d950a897],
    [0x3c4363ed60c2ac11, 0x3feece086061892d],
    [0x3c9666093b0664ef, 0x3feeca41ed1d0057],
    [0x3c6ecce1daa10379, 0x3feec6a2b5c13cd0],
    [0x3c93ff8e3f0f1230, 0x3feec32af0d7d3de],
    [0x3c7690cebb7aafb0, 0x3feebfdad5362a27],
    [0x3c931dbdeb54e077, 0x3feebcb299fddd0d],
    [0xbc8f94340071a38e, 0x3feeb9b2769d2ca7],
    [0xbc87deccdc93a349, 0x3feeb6daa2cf6642],
    [0xbc78dec6bd0f385f, 0x3feeb42b569d4f82],
    [0xbc861246ec7b5cf6, 0x3feeb1a4ca5d920f],
    [0x3c93350518fdd78e, 0x3feeaf4736b527da],
    [0x3c7b98b72f8a9b05, 0x3feead12d497c7fd],
    [0x3c9063e1e21c5409, 0x3feeab07dd485429],
    [0x3c34c7855019c6ea, 0x3feea9268a5946b7],
    [0x3c9432e62b64c035, 0x3feea76f15ad2148],
    [0xbc8ce44a6199769f, 0x3feea5e1b976dc09],
    [0xbc8c33c53bef4da8, 0x3feea47eb03a5585],
    [0xbc845378892be9ae, 0x3feea34634ccc320],
    [0xbc93cedd78565858, 0x3feea23882552225],
    [0x3c5710aa807e1964, 0x3feea155d44ca973],
    [0xbc93b3efbf5e2228, 0x3feea09e667f3bcd],
    [0xbc6a12ad8734b982, 0x3feea012750bdabf],
    [0xbc6367efb86da9ee, 0x3fee9fb23c651a2f],
    [0xbc80dc3d54e08851, 0x3fee9f7df9519484],
    [0xbc781f647e5a3ecf, 0x3fee9f75e8ec5f74],
    [0xbc86ee4ac08b7db0, 0x3fee9f9a48a58174],
    [0xbc8619321e55e68a, 0x3fee9feb564267c9],
    [0x3c909ccb5e09d4d3, 0x3feea0694fde5d3f],
    [0xbc7b32dcb94da51d, 0x3feea11473eb0187],
    [0x3c94ecfd5467c06b, 0x3feea1ed0130c132],
    [0x3c65ebe1abd66c55, 0x3feea2f336cf4e62],
    [0xbc88a1c52fb3cf42, 0x3feea427543e1a12],
    [0xbc9369b6f13b3734, 0x3feea589994cce13],
    [0xbc805e843a19ff1e, 0x3feea71a4623c7ad],
    [0xbc94d450d872576e, 0x3feea8d99b4492ed],
    [0x3c90ad675b0e8a00, 0x3feeaac7d98a6699],
    [0x3c8db72fc1f0eab4, 0x3feeace5422aa0db],
    [0xbc65b6609cc5e7ff, 0x3feeaf3216b5448c],
    [0x3c7bf68359f35f44, 0x3feeb1ae99157736],
    [0xbc93091fa71e3d83, 0x3feeb45b0b91ffc6],
    [0xbc5da9b88b6c1e29, 0x3feeb737b0cdc5e5],
    [0xbc6c23f97c90b959, 0x3feeba44cbc8520f],
    [0xbc92434322f4f9aa, 0x3feebd829fde4e50],
    [0xbc85ca6cd7668e4b, 0x3feec0f170ca07ba],
    [0x3c71affc2b91ce27, 0x3feec49182a3f090],
    [0x3c6dd235e10a73bb, 0x3feec86319e32323],
    [0xbc87c50422622263, 0x3feecc667b5de565],
    [0x3c8b1c86e3e231d5, 0x3feed09bec4a2d33],
    [0xbc91bbd1d3bcbb15, 0x3feed503b23e255d],
    [0x3c90cc319cee31d2, 0x3feed99e1330b358],
    [0x3c8469846e735ab3, 0x3feede6b5579fdbf],
    [0xbc82dfcd978e9db4, 0x3feee36bbfd3f37a],
    [0x3c8c1a7792cb3387, 0x3feee89f995ad3ad],
    [0xbc907b8f4ad1d9fa, 0x3feeee07298db666],
    [0xbc55c3d956dcaeba, 0x3feef3a2b84f15fb],
    [0xbc90a40e3da6f640, 0x3feef9728de5593a],
    [0xbc68d6f438ad9334, 0x3feeff76f2fb5e47],
    [0xbc91eee26b588a35, 0x3fef05b030a1064a],
    [0x3c74ffd70a5fddcd, 0x3fef0c1e904bc1d2],
    [0xbc91bdfbfa9298ac, 0x3fef12c25bd71e09],
    [0x3c736eae30af0cb3, 0x3fef199bdd85529c],
    [0x3c8ee3325c9ffd94, 0x3fef20ab5fffd07a],
    [0x3c84e08fd10959ac, 0x3fef27f12e57d14b],
    [0x3c63cdaf384e1a67, 0x3fef2f6d9406e7b5],
    [0x3c676b2c6c921968, 0x3fef3720dcef9069],
    [0xbc808a1883ccb5d2, 0x3fef3f0b555dc3fa],
    [0xbc8fad5d3ffffa6f, 0x3fef472d4a07897c],
    [0xbc900dae3875a949, 0x3fef4f87080d89f2],
    [0x3c74a385a63d07a7, 0x3fef5818dcfba487],
    [0xbc82919e2040220f, 0x3fef60e316c98398],
    [0x3c8e5a50d5c192ac, 0x3fef69e603db3285],
    [0x3c843a59ac016b4b, 0x3fef7321f301b460],
    [0xbc82d52107b43e1f, 0x3fef7c97337b9b5f],
    [0xbc892ab93b470dc9, 0x3fef864614f5a129],
    [0x3c74b604603a88d3, 0x3fef902ee78b3ff6],
    [0x3c83c5ec519d7271, 0x3fef9a51fbc74c83],
    [0xbc8ff7128fd391f0, 0x3fefa4afa2a490da],
    [0xbc8dae98e223747d, 0x3fefaf482d8e67f1],
    [0x3c8ec3bc41aa2008, 0x3fefba1bee615a27],
    [0x3c842b94c3a9eb32, 0x3fefc52b376bba97],
    [0x3c8a64a931d185ee, 0x3fefd0765b6e4540],
    [0xbc8e37bae43be3ed, 0x3fefdbfdad9cbe14],
    [0x3c77893b4d91cd9d, 0x3fefe7c1819e90d8],
    [0x3c5305c14160cc89, 0x3feff3c22b8f71f1],
];

fn top12(x: f64) -> u32 {
    (x.to_bits() >> 52) as u32
}

pub fn exp(x: f64) -> f64 {
    let mut abstop = top12(x) & 0x7ff;
    // Outside 2^-54 <= |x| < 512, `top12` of each.
    if abstop.wrapping_sub(0x3c9) >= 0x408 - 0x3c9 {
        if abstop.wrapping_sub(0x3c9) >= 0x8000_0000 {
            return 1.0 + x;
        }
        if abstop >= 0x409 {
            if x == f64::NEG_INFINITY {
                return 0.0;
            }
            if abstop >= 0x7ff {
                return 1.0 + x;
            }
            return if x.is_sign_negative() {
                0.0
            } else {
                f64::INFINITY
            };
        }
        abstop = 0;
    }
    let kd = INV_LN2_N * x + SHIFT;
    let ki = kd.to_bits();
    let kd = kd - SHIFT;
    let r = x + kd * NEG_LN2_HI_N + kd * NEG_LN2_LO_N;
    let [tail, scale] = T[(ki % N) as usize];
    let tail = f64::from_bits(tail);
    let sbits = scale.wrapping_add(ki << 45);
    let r2 = r * r;
    let tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
    if abstop == 0 {
        return scaled_far(tmp, sbits, ki);
    }
    let scale = f64::from_bits(sbits);
    scale + scale * tmp
}

/// `scale` itself would leave the exponent's range: scaled back in by 2^1009 above, and below
/// rounded once into the subnormals.
fn scaled_far(tmp: f64, sbits: u64, ki: u64) -> f64 {
    if ki & 0x8000_0000 == 0 {
        let scale = f64::from_bits(sbits.wrapping_sub(1009 << 52));
        return f64::from_bits(0x7f00000000000000) * (scale + scale * tmp);
    }
    let scale = f64::from_bits(sbits.wrapping_add(1022 << 52));
    let mut y = scale + scale * tmp;
    if y < 1.0 {
        let lo = scale - y + scale * tmp;
        let hi = 1.0 + y;
        let lo = 1.0 - hi + y + lo;
        // `+ 0.0` turns a -0 from the subtraction into +0.
        y = (hi + lo) - 1.0 + 0.0;
    }
    f64::from_bits(0x0010000000000000) * y
}