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

↓

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

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 r223844 = 1.0;
        double r223845 = 4.0;
        double r223846 = x;
        double r223847 = y;
        double r223848 = 0.25;
        double r223849 = r223847 * r223848;
        double r223850 = r223846 + r223849;
        double r223851 = z;
        double r223852 = r223850 - r223851;
        double r223853 = r223845 * r223852;
        double r223854 = r223853 / r223847;
        double r223855 = r223844 + r223854;
        return r223855;
}

↓

double f(double x, double y, double z) {
        double r223856 = 4.0;
        double r223857 = x;
        double r223858 = y;
        double r223859 = r223857 / r223858;
        double r223860 = 2.0;
        double r223861 = z;
        double r223862 = r223861 / r223858;
        double r223863 = r223856 * r223862;
        double r223864 = r223860 - r223863;
        double r223865 = fma(r223856, r223859, r223864);
        return r223865;
}

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 2020001 +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