micromath 2.0.0

//! x^n where n is an integer.
//!
//! Method described at: <https://stackoverflow.com/a/101613>
//!
//! Note: not optimal for all powers, is NP complete
//! <https://en.wikipedia.org/wiki/Addition-chain_exponentiation>
//! will result in an addition multiplication for x^15 for example

use super::{F32, MANTISSA_BITS};

impl F32 {
    /// Approximates a number raised to an integer power.
    pub fn powi(self, n: i32) -> Self {
        let mut base = self;
        let mut abs_n = i32::abs(n);
        let mut result = 1.0;

        if n < 0 {
            base = 1.0 / self;
        }

        if n == 0 {
            return Self::ONE;
        }

        // 0.0 == 0.0 and -0.0 according to IEEE standards.
        if self == Self::ZERO && n > 0 {
            return self;
        }

        // For values less than 2.0, but greater than 0.5 (1.0/2.0), you can multiply longer without
        // going over exponent, i.e. 1.1 multiplied against itself will grow slowly.
        if !(0.5..2.0).contains(&self.abs().0) {
            // Approximation if we end up outside of the range of floating point values,
            // then we end early
            let approx_final_exponent = self.extract_exponent_value() * n;
            let max_representable_exponent = 127;
            let min_representable_exponent = -126 - (MANTISSA_BITS as i32);
            if approx_final_exponent > max_representable_exponent
                || (self == Self::ZERO && approx_final_exponent < 0)
            {
                if self.is_sign_positive() || n & 1 == 0 {
                    return Self::INFINITY;
                } else {
                    return Self::NEG_INFINITY;
                }
            } else if approx_final_exponent < min_representable_exponent {
                // We may want to copy the sign and do the same thing as above,
                // but that seems like an awful amount of work when 99.99999% of people only care
                // about bare zero
                return Self::ZERO;
            }
        }

        loop {
            if (abs_n & 1) == 1 {
                result *= base;
            }

            abs_n >>= 1;

            if abs_n == 0 {
                return Self(result);
            }

            base *= base;
        }
    }
}

#[cfg(test)]
mod tests {
    use super::F32;

    /// Error builds up from both exp and ln approximation, so we double the error allowed.
    pub(crate) const MAX_ERROR: f32 = 0.002;

    /// `powi(3,x)` test vectors - `(input_base, input_power, output)`
    pub(crate) const TEST_VECTORS: &[(f32, f32, f32)] = &[
        (-100000.0, -128.0, 0.0),
        (-10.0, -128.0, 0.0),
        (-3.0, -128.0, 0.0),
        (-2.0, -128.0, 2.938736e-39),
        (-1.0, -128.0, 1.0),
        (1.0, -128.0, 1.0),
        (2.0, -128.0, 2.938736e-39),
        (3.0, -128.0, 0.0),
        (10.0, -128.0, 0.0),
        (100000.0, -128.0, 0.0),
        (-100000.0, -127.0, -0.0),
        (-10.0, -127.0, -0.0),
        (-3.0, -127.0, -0.0),
        (-2.0, -127.0, -5.877472e-39),
        (-1.0, -127.0, -1.0),
        (1.0, -127.0, 1.0),
        (2.0, -127.0, 5.877472e-39),
        (3.0, -127.0, 0.0),
        (10.0, -127.0, 0.0),
        (100000.0, -127.0, 0.0),
        (-100000.0, -126.0, 0.0),
        (-10.0, -126.0, 0.0),
        (-3.0, -126.0, 0.0),
        (-2.0, -126.0, 1.1754944e-38),
        (-1.0, -126.0, 1.0),
        (1.0, -126.0, 1.0),
        (2.0, -126.0, 1.1754944e-38),
        (3.0, -126.0, 0.0),
        (10.0, -126.0, 0.0),
        (100000.0, -126.0, 0.0),
        (-100000.0, -125.0, -0.0),
        (-10.0, -125.0, -0.0),
        (-3.0, -125.0, -0.0),
        (-2.0, -125.0, -2.3509887e-38),
        (-1.0, -125.0, -1.0),
        (1.0, -125.0, 1.0),
        (2.0, -125.0, 2.3509887e-38),
        (3.0, -125.0, 0.0),
        (10.0, -125.0, 0.0),
        (100000.0, -125.0, 0.0),
        (-100000.0, -124.0, 0.0),
        (-10.0, -124.0, 0.0),
        (-3.0, -124.0, 0.0),
        (-2.0, -124.0, 4.7019774e-38),
        (-1.0, -124.0, 1.0),
        (1.0, -124.0, 1.0),
        (2.0, -124.0, 4.7019774e-38),
        (3.0, -124.0, 0.0),
        (10.0, -124.0, 0.0),
        (100000.0, -124.0, 0.0),
        (-100000.0, -123.0, -0.0),
        (-10.0, -123.0, -0.0),
        (-3.0, -123.0, -0.0),
        (-2.0, -123.0, -9.403955e-38),
        (-1.0, -123.0, -1.0),
        (1.0, -123.0, 1.0),
        (2.0, -123.0, 9.403955e-38),
        (3.0, -123.0, 0.0),
        (10.0, -123.0, 0.0),
        (100000.0, -123.0, 0.0),
        (-100000.0, -122.0, 0.0),
        (-10.0, -122.0, 0.0),
        (-3.0, -122.0, 0.0),
        (-2.0, -122.0, 1.880791e-37),
        (-1.0, -122.0, 1.0),
        (1.0, -122.0, 1.0),
        (2.0, -122.0, 1.880791e-37),
        (3.0, -122.0, 0.0),
        (10.0, -122.0, 0.0),
        (100000.0, -122.0, 0.0),
        (-100000.0, -121.0, -0.0),
        (-10.0, -121.0, -0.0),
        (-3.0, -121.0, -0.0),
        (-2.0, -121.0, -3.761582e-37),
        (-1.0, -121.0, -1.0),
        (1.0, -121.0, 1.0),
        (2.0, -121.0, 3.761582e-37),
        (3.0, -121.0, 0.0),
        (10.0, -121.0, 0.0),
        (100000.0, -121.0, 0.0),
        (-100000.0, -120.0, 0.0),
        (-10.0, -120.0, 0.0),
        (-3.0, -120.0, 0.0),
        (-2.0, -120.0, 7.523164e-37),
        (-1.0, -120.0, 1.0),
        (1.0, -120.0, 1.0),
        (2.0, -120.0, 7.523164e-37),
        (3.0, -120.0, 0.0),
        (10.0, -120.0, 0.0),
        (100000.0, -120.0, 0.0),
        (-100000.0, -119.0, -0.0),
        (-10.0, -119.0, -0.0),
        (-3.0, -119.0, -0.0),
        (-2.0, -119.0, -1.5046328e-36),
        (-1.0, -119.0, -1.0),
        (1.0, -119.0, 1.0),
        (2.0, -119.0, 1.5046328e-36),
        (3.0, -119.0, 0.0),
        (10.0, -119.0, 0.0),
        (100000.0, -119.0, 0.0),
        (-100000.0, -118.0, 0.0),
        (-10.0, -118.0, 0.0),
        (-3.0, -118.0, 0.0),
        (-2.0, -118.0, 3.0092655e-36),
        (-1.0, -118.0, 1.0),
        (1.0, -118.0, 1.0),
        (2.0, -118.0, 3.0092655e-36),
        (3.0, -118.0, 0.0),
        (10.0, -118.0, 0.0),
        (100000.0, -118.0, 0.0),
        (-100000.0, -117.0, -0.0),
        (-10.0, -117.0, -0.0),
        (-3.0, -117.0, -0.0),
        (-2.0, -117.0, -6.018531e-36),
        (-1.0, -117.0, -1.0),
        (1.0, -117.0, 1.0),
        (2.0, -117.0, 6.018531e-36),
        (3.0, -117.0, 0.0),
        (10.0, -117.0, 0.0),
        (100000.0, -117.0, 0.0),
        (-100000.0, -116.0, 0.0),
        (-10.0, -116.0, 0.0),
        (-3.0, -116.0, 0.0),
        (-2.0, -116.0, 1.2037062e-35),
        (-1.0, -116.0, 1.0),
        (1.0, -116.0, 1.0),
        (2.0, -116.0, 1.2037062e-35),
        (3.0, -116.0, 0.0),
        (10.0, -116.0, 0.0),
        (100000.0, -116.0, 0.0),
        (-100000.0, -115.0, -0.0),
        (-10.0, -115.0, -0.0),
        (-3.0, -115.0, -0.0),
        (-2.0, -115.0, -2.4074124e-35),
        (-1.0, -115.0, -1.0),
        (1.0, -115.0, 1.0),
        (2.0, -115.0, 2.4074124e-35),
        (3.0, -115.0, 0.0),
        (10.0, -115.0, 0.0),
        (100000.0, -115.0, 0.0),
        (-100000.0, -114.0, 0.0),
        (-10.0, -114.0, 0.0),
        (-3.0, -114.0, 0.0),
        (-2.0, -114.0, 4.814825e-35),
        (-1.0, -114.0, 1.0),
        (1.0, -114.0, 1.0),
        (2.0, -114.0, 4.814825e-35),
        (3.0, -114.0, 0.0),
        (10.0, -114.0, 0.0),
        (100000.0, -114.0, 0.0),
        (-100000.0, -113.0, -0.0),
        (-10.0, -113.0, -0.0),
        (-3.0, -113.0, -0.0),
        (-2.0, -113.0, -9.62965e-35),
        (-1.0, -113.0, -1.0),
        (1.0, -113.0, 1.0),
        (2.0, -113.0, 9.62965e-35),
        (3.0, -113.0, 0.0),
        (10.0, -113.0, 0.0),
        (100000.0, -113.0, 0.0),
        (-100000.0, -112.0, 0.0),
        (-10.0, -112.0, 0.0),
        (-3.0, -112.0, 0.0),
        (-2.0, -112.0, 1.92593e-34),
        (-1.0, -112.0, 1.0),
        (1.0, -112.0, 1.0),
        (2.0, -112.0, 1.92593e-34),
        (3.0, -112.0, 0.0),
        (10.0, -112.0, 0.0),
        (100000.0, -112.0, 0.0),
        (-100000.0, -111.0, -0.0),
        (-10.0, -111.0, -0.0),
        (-3.0, -111.0, -0.0),
        (-2.0, -111.0, -3.85186e-34),
        (-1.0, -111.0, -1.0),
        (1.0, -111.0, 1.0),
        (2.0, -111.0, 3.85186e-34),
        (3.0, -111.0, 0.0),
        (10.0, -111.0, 0.0),
        (100000.0, -111.0, 0.0),
        (-100000.0, -110.0, 0.0),
        (-10.0, -110.0, 0.0),
        (-3.0, -110.0, 0.0),
        (-2.0, -110.0, 7.70372e-34),
        (-1.0, -110.0, 1.0),
        (1.0, -110.0, 1.0),
        (2.0, -110.0, 7.70372e-34),
        (3.0, -110.0, 0.0),
        (10.0, -110.0, 0.0),
        (100000.0, -110.0, 0.0),
        (-100000.0, -109.0, -0.0),
        (-10.0, -109.0, -0.0),
        (-3.0, -109.0, -0.0),
        (-2.0, -109.0, -1.540744e-33),
        (-1.0, -109.0, -1.0),
        (1.0, -109.0, 1.0),
        (2.0, -109.0, 1.540744e-33),
        (3.0, -109.0, 0.0),
        (10.0, -109.0, 0.0),
        (100000.0, -109.0, 0.0),
        (-100000.0, -108.0, 0.0),
        (-10.0, -108.0, 0.0),
        (-3.0, -108.0, 0.0),
        (-2.0, -108.0, 3.081488e-33),
        (-1.0, -108.0, 1.0),
        (1.0, -108.0, 1.0),
        (2.0, -108.0, 3.081488e-33),
        (3.0, -108.0, 0.0),
        (10.0, -108.0, 0.0),
        (100000.0, -108.0, 0.0),
        (-100000.0, -107.0, -0.0),
        (-10.0, -107.0, -0.0),
        (-3.0, -107.0, -0.0),
        (-2.0, -107.0, -6.162976e-33),
        (-1.0, -107.0, -1.0),
        (1.0, -107.0, 1.0),
        (2.0, -107.0, 6.162976e-33),
        (3.0, -107.0, 0.0),
        (10.0, -107.0, 0.0),
        (100000.0, -107.0, 0.0),
        (-100000.0, -106.0, 0.0),
        (-10.0, -106.0, 0.0),
        (-3.0, -106.0, 0.0),
        (-2.0, -106.0, 1.2325952e-32),
        (-1.0, -106.0, 1.0),
        (1.0, -106.0, 1.0),
        (2.0, -106.0, 1.2325952e-32),
        (3.0, -106.0, 0.0),
        (10.0, -106.0, 0.0),
        (100000.0, -106.0, 0.0),
        (-100000.0, -105.0, -0.0),
        (-10.0, -105.0, -0.0),
        (-3.0, -105.0, -0.0),
        (-2.0, -105.0, -2.4651903e-32),
        (-1.0, -105.0, -1.0),
        (1.0, -105.0, 1.0),
        (2.0, -105.0, 2.4651903e-32),
        (3.0, -105.0, 0.0),
        (10.0, -105.0, 0.0),
        (100000.0, -105.0, 0.0),
        (-100000.0, -104.0, 0.0),
        (-10.0, -104.0, 0.0),
        (-3.0, -104.0, 0.0),
        (-2.0, -104.0, 4.9303807e-32),
        (-1.0, -104.0, 1.0),
        (1.0, -104.0, 1.0),
        (2.0, -104.0, 4.9303807e-32),
        (3.0, -104.0, 0.0),
        (10.0, -104.0, 0.0),
        (100000.0, -104.0, 0.0),
        (-100000.0, -103.0, -0.0),
        (-10.0, -103.0, -0.0),
        (-3.0, -103.0, -0.0),
        (-2.0, -103.0, -9.8607613e-32),
        (-1.0, -103.0, -1.0),
        (1.0, -103.0, 1.0),
        (2.0, -103.0, 9.8607613e-32),
        (3.0, -103.0, 0.0),
        (10.0, -103.0, 0.0),
        (100000.0, -103.0, 0.0),
        (-100000.0, -102.0, 0.0),
        (-10.0, -102.0, 0.0),
        (-3.0, -102.0, 0.0),
        (-2.0, -102.0, 1.9721523e-31),
        (-1.0, -102.0, 1.0),
        (1.0, -102.0, 1.0),
        (2.0, -102.0, 1.9721523e-31),
        (3.0, -102.0, 0.0),
        (10.0, -102.0, 0.0),
        (100000.0, -102.0, 0.0),
        (-100000.0, -101.0, -0.0),
        (-10.0, -101.0, -0.0),
        (-3.0, -101.0, -0.0),
        (-2.0, -101.0, -3.9443045e-31),
        (-1.0, -101.0, -1.0),
        (1.0, -101.0, 1.0),
        (2.0, -101.0, 3.9443045e-31),
        (3.0, -101.0, 0.0),
        (10.0, -101.0, 0.0),
        (100000.0, -101.0, 0.0),
        (-100000.0, -100.0, 0.0),
        (-10.0, -100.0, 0.0),
        (-3.0, -100.0, 0.0),
        (-2.0, -100.0, 7.888609e-31),
        (-1.0, -100.0, 1.0),
        (1.0, -100.0, 1.0),
        (2.0, -100.0, 7.888609e-31),
        (3.0, -100.0, 0.0),
        (10.0, -100.0, 0.0),
        (100000.0, -100.0, 0.0),
        (-100000.0, -99.0, -0.0),
        (-10.0, -99.0, -0.0),
        (-3.0, -99.0, -0.0),
        (-2.0, -99.0, -1.5777218e-30),
        (-1.0, -99.0, -1.0),
        (1.0, -99.0, 1.0),
        (2.0, -99.0, 1.5777218e-30),
        (3.0, -99.0, 0.0),
        (10.0, -99.0, 0.0),
        (100000.0, -99.0, 0.0),
        (-100000.0, -98.0, 0.0),
        (-10.0, -98.0, 0.0),
        (-3.0, -98.0, 0.0),
        (-2.0, -98.0, 3.1554436e-30),
        (-1.0, -98.0, 1.0),
        (1.0, -98.0, 1.0),
        (2.0, -98.0, 3.1554436e-30),
        (3.0, -98.0, 0.0),
        (10.0, -98.0, 0.0),
        (100000.0, -98.0, 0.0),
        (-100000.0, -97.0, -0.0),
        (-10.0, -97.0, -0.0),
        (-3.0, -97.0, -0.0),
        (-2.0, -97.0, -6.3108872e-30),
        (-1.0, -97.0, -1.0),
        (1.0, -97.0, 1.0),
        (2.0, -97.0, 6.3108872e-30),
        (3.0, -97.0, 0.0),
        (10.0, -97.0, 0.0),
        (100000.0, -97.0, 0.0),
        (-100000.0, -96.0, 0.0),
        (-10.0, -96.0, 0.0),
        (-3.0, -96.0, 0.0),
        (-2.0, -96.0, 1.2621775e-29),
        (-1.0, -96.0, 1.0),
        (1.0, -96.0, 1.0),
        (2.0, -96.0, 1.2621775e-29),
        (3.0, -96.0, 0.0),
        (10.0, -96.0, 0.0),
        (100000.0, -96.0, 0.0),
        (-100000.0, -95.0, -0.0),
        (-10.0, -95.0, -0.0),
        (-3.0, -95.0, -0.0),
        (-2.0, -95.0, -2.524355e-29),
        (-1.0, -95.0, -1.0),
        (1.0, -95.0, 1.0),
        (2.0, -95.0, 2.524355e-29),
        (3.0, -95.0, 0.0),
        (10.0, -95.0, 0.0),
        (100000.0, -95.0, 0.0),
        (-100000.0, -94.0, 0.0),
        (-10.0, -94.0, 0.0),
        (-3.0, -94.0, 1e-45),
        (-2.0, -94.0, 5.04871e-29),
        (-1.0, -94.0, 1.0),
        (1.0, -94.0, 1.0),
        (2.0, -94.0, 5.04871e-29),
        (3.0, -94.0, 1e-45),
        (10.0, -94.0, 0.0),
        (100000.0, -94.0, 0.0),
        (-100000.0, -93.0, -0.0),
        (-10.0, -93.0, -0.0),
        (-3.0, -93.0, -4e-45),
        (-2.0, -93.0, -1.009742e-28),
        (-1.0, -93.0, -1.0),
        (1.0, -93.0, 1.0),
        (2.0, -93.0, 1.009742e-28),
        (3.0, -93.0, 4e-45),
        (10.0, -93.0, 0.0),
        (100000.0, -93.0, 0.0),
        (-100000.0, -92.0, 0.0),
        (-10.0, -92.0, 0.0),
        (-3.0, -92.0, 1.3e-44),
        (-2.0, -92.0, 2.019484e-28),
        (-1.0, -92.0, 1.0),
        (1.0, -92.0, 1.0),
        (2.0, -92.0, 2.019484e-28),
        (3.0, -92.0, 1.3e-44),
        (10.0, -92.0, 0.0),
        (100000.0, -92.0, 0.0),
        (-100000.0, -91.0, -0.0),
        (-10.0, -91.0, -0.0),
        (-3.0, -91.0, -3.8e-44),
        (-2.0, -91.0, -4.038968e-28),
        (-1.0, -91.0, -1.0),
        (1.0, -91.0, 1.0),
        (2.0, -91.0, 4.038968e-28),
        (3.0, -91.0, 3.8e-44),
        (10.0, -91.0, 0.0),
        (100000.0, -91.0, 0.0),
        (-100000.0, -90.0, 0.0),
        (-10.0, -90.0, 0.0),
        (-3.0, -90.0, 1.15e-43),
        (-2.0, -90.0, 8.077936e-28),
        (-1.0, -90.0, 1.0),
        (1.0, -90.0, 1.0),
        (2.0, -90.0, 8.077936e-28),
        (3.0, -90.0, 1.15e-43),
        (10.0, -90.0, 0.0),
        (100000.0, -90.0, 0.0),
        (-100000.0, -89.0, -0.0),
        (-10.0, -89.0, -0.0),
        (-3.0, -89.0, -3.43e-43),
        (-2.0, -89.0, -1.6155871e-27),
        (-1.0, -89.0, -1.0),
        (1.0, -89.0, 1.0),
        (2.0, -89.0, 1.6155871e-27),
        (3.0, -89.0, 3.43e-43),
        (10.0, -89.0, 0.0),
        (100000.0, -89.0, 0.0),
        (-100000.0, -88.0, 0.0),
        (-10.0, -88.0, 0.0),
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        (-10.0, 126.0, f32::INFINITY),
        (-3.0, 126.0, f32::INFINITY),
        (-2.0, 126.0, 8.507_059e37),
        (-1.0, 126.0, 1.0),
        (1.0, 126.0, 1.0),
        (2.0, 126.0, 8.507_059e37),
        (3.0, 126.0, f32::INFINITY),
        (10.0, 126.0, f32::INFINITY),
        (100000.0, 126.0, f32::INFINITY),
        (-100000.0, 127.0, -f32::INFINITY),
        (-10.0, 127.0, -f32::INFINITY),
        (-3.0, 127.0, -f32::INFINITY),
        (-2.0, 127.0, -1.701_411_8e38),
        (-1.0, 127.0, -1.0),
        (1.0, 127.0, 1.0),
        (2.0, 127.0, 1.701_411_8e38),
        (3.0, 127.0, f32::INFINITY),
        (10.0, 127.0, f32::INFINITY),
        (100000.0, 127.0, f32::INFINITY),
        (-100000.0, 128.0, f32::INFINITY),
        (-10.0, 128.0, f32::INFINITY),
        (-3.0, 128.0, f32::INFINITY),
        (-2.0, 128.0, f32::INFINITY),
        (-1.0, 128.0, 1.0),
        (1.0, 128.0, 1.0),
        (2.0, 128.0, f32::INFINITY),
        (3.0, 128.0, f32::INFINITY),
        (10.0, 128.0, f32::INFINITY),
        (100000.0, 128.0, f32::INFINITY),
        (0.0, 3.0, 0.0),
        (-0.0, 3.0, 0.0),
        (0.0, -3.0, f32::INFINITY),
        (-0.0, -3.0, -f32::INFINITY),
        (-0.0, -2.0, f32::INFINITY),
        (0.75, -150.0, 5.505_674e18),
        (-0.75, -150.0, 5.505_674e18),
        (1.1, -150.0, 6.181_527_4e-7),
        (1.1, 150.0, 1617723.1),
    ];

    fn calc_relative_error(experimental: F32, expected: f32) -> F32 {
        if experimental.is_nan() && expected.is_nan() || experimental.0 == expected {
            F32::ZERO
        } else if expected != 0.0 {
            (experimental - expected).abs() / expected
        } else {
            (experimental - expected).abs() / (expected + 1.0e-20)
        }
    }

    #[test]
    fn sanity_check() {
        for &(base_input, power_input, expected) in TEST_VECTORS {
            let exp_x = F32(base_input).powi(power_input as i32);
            let relative_error = calc_relative_error(exp_x, expected);

            assert!(
                relative_error <= MAX_ERROR,
                "relative_error {} too large for input {}.powi({}) : {} vs {}",
                relative_error,
                base_input,
                power_input,
                exp_x,
                expected
            );
        }
    }
}