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

↓

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

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 r199702 = 1.0;
        double r199703 = 4.0;
        double r199704 = x;
        double r199705 = y;
        double r199706 = 0.25;
        double r199707 = r199705 * r199706;
        double r199708 = r199704 + r199707;
        double r199709 = z;
        double r199710 = r199708 - r199709;
        double r199711 = r199703 * r199710;
        double r199712 = r199711 / r199705;
        double r199713 = r199702 + r199712;
        return r199713;
}

↓

double f(double x, double y, double z) {
        double r199714 = 4.0;
        double r199715 = x;
        double r199716 = y;
        double r199717 = r199715 / r199716;
        double r199718 = 2.0;
        double r199719 = z;
        double r199720 = r199719 / r199716;
        double r199721 = r199714 * r199720;
        double r199722 = r199718 - r199721;
        double r199723 = fma(r199714, r199717, r199722);
        return r199723;
}

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