\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

↓

\[\mathsf{fma}\left(4, \frac{x}{y}, 2 - 4 \cdot \frac{z}{y}\right)\]

1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}

↓

\mathsf{fma}\left(4, \frac{x}{y}, 2 - 4 \cdot \frac{z}{y}\right)

double f(double x, double y, double z) {
        double r232938 = 1.0;
        double r232939 = 4.0;
        double r232940 = x;
        double r232941 = y;
        double r232942 = 0.25;
        double r232943 = r232941 * r232942;
        double r232944 = r232940 + r232943;
        double r232945 = z;
        double r232946 = r232944 - r232945;
        double r232947 = r232939 * r232946;
        double r232948 = r232947 / r232941;
        double r232949 = r232938 + r232948;
        return r232949;
}

↓

double f(double x, double y, double z) {
        double r232950 = 4.0;
        double r232951 = x;
        double r232952 = y;
        double r232953 = r232951 / r232952;
        double r232954 = 2.0;
        double r232955 = z;
        double r232956 = r232955 / r232952;
        double r232957 = r232950 * r232956;
        double r232958 = r232954 - r232957;
        double r232959 = fma(r232950, r232953, r232958);
        return r232959;
}

Derivation

Initial program 0.1
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
Taylor expanded around 0 0.0
\[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{y}, 2 - 4 \cdot \frac{z}{y}\right)}\]
Final simplification0.0
\[\leadsto \mathsf{fma}\left(4, \frac{x}{y}, 2 - 4 \cdot \frac{z}{y}\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))

Error

Derivation

Reproduce