Average Error: 0.2 → 0.0
Time: 18.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(0.75 + \left(\frac{x}{y} - \frac{z}{y}\right), 4, 1\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(0.75 + \left(\frac{x}{y} - \frac{z}{y}\right), 4, 1\right)
double f(double x, double y, double z) {
        double r166791 = 1.0;
        double r166792 = 4.0;
        double r166793 = x;
        double r166794 = y;
        double r166795 = 0.75;
        double r166796 = r166794 * r166795;
        double r166797 = r166793 + r166796;
        double r166798 = z;
        double r166799 = r166797 - r166798;
        double r166800 = r166792 * r166799;
        double r166801 = r166800 / r166794;
        double r166802 = r166791 + r166801;
        return r166802;
}
↓
double f(double x, double y, double z) {
        double r166803 = 0.75;
        double r166804 = x;
        double r166805 = y;
        double r166806 = r166804 / r166805;
        double r166807 = z;
        double r166808 = r166807 / r166805;
        double r166809 = r166806 - r166808;
        double r166810 = r166803 + r166809;
        double r166811 = 4.0;
        double r166812 = 1.0;
        double r166813 = fma(r166810, r166811, r166812);
        return r166813;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75 + \frac{x - z}{y}, 4, 1\right)}\]
  3. Using strategy rm

  4. Applied div-sub0.0

    \[\leadsto \mathsf{fma}\left(0.75 + \color{blue}{\left(\frac{x}{y} - \frac{z}{y}\right)}, 4, 1\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.75 + \left(\frac{x}{y} - \frac{z}{y}\right), 4, 1\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))