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

↓

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

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 r204743 = 1.0;
        double r204744 = 4.0;
        double r204745 = x;
        double r204746 = y;
        double r204747 = 0.25;
        double r204748 = r204746 * r204747;
        double r204749 = r204745 + r204748;
        double r204750 = z;
        double r204751 = r204749 - r204750;
        double r204752 = r204744 * r204751;
        double r204753 = r204752 / r204746;
        double r204754 = r204743 + r204753;
        return r204754;
}

↓

double f(double x, double y, double z) {
        double r204755 = 4.0;
        double r204756 = x;
        double r204757 = y;
        double r204758 = r204756 / r204757;
        double r204759 = 2.0;
        double r204760 = z;
        double r204761 = r204760 / r204757;
        double r204762 = r204755 * r204761;
        double r204763 = r204759 - r204762;
        double r204764 = fma(r204755, r204758, r204763);
        return r204764;
}

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 2020036 +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