Average Error: 0.2 → 0.0
Time: 17.7s
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 r180921 = 1.0;
        double r180922 = 4.0;
        double r180923 = x;
        double r180924 = y;
        double r180925 = 0.75;
        double r180926 = r180924 * r180925;
        double r180927 = r180923 + r180926;
        double r180928 = z;
        double r180929 = r180927 - r180928;
        double r180930 = r180922 * r180929;
        double r180931 = r180930 / r180924;
        double r180932 = r180921 + r180931;
        return r180932;
}
↓
double f(double x, double y, double z) {
        double r180933 = 0.75;
        double r180934 = x;
        double r180935 = y;
        double r180936 = r180934 / r180935;
        double r180937 = z;
        double r180938 = r180937 / r180935;
        double r180939 = r180936 - r180938;
        double r180940 = r180933 + r180939;
        double r180941 = 4.0;
        double r180942 = 1.0;
        double r180943 = fma(r180940, r180941, r180942);
        return r180943;
}

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)))