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

↓

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

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 r254752 = 1.0;
        double r254753 = 4.0;
        double r254754 = x;
        double r254755 = y;
        double r254756 = 0.25;
        double r254757 = r254755 * r254756;
        double r254758 = r254754 + r254757;
        double r254759 = z;
        double r254760 = r254758 - r254759;
        double r254761 = r254753 * r254760;
        double r254762 = r254761 / r254755;
        double r254763 = r254752 + r254762;
        return r254763;
}

↓

double f(double x, double y, double z) {
        double r254764 = 4.0;
        double r254765 = x;
        double r254766 = y;
        double r254767 = r254765 / r254766;
        double r254768 = 2.0;
        double r254769 = z;
        double r254770 = r254769 / r254766;
        double r254771 = r254764 * r254770;
        double r254772 = r254768 - r254771;
        double r254773 = fma(r254764, r254767, r254772);
        return r254773;
}

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 2020027 +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