Average Error: 0.3 → 0.2
Time: 1.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{4}{y}, x - z, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{4}{y}, x - z, 4\right)
double f(double x, double y, double z) {
        double r251789 = 1.0;
        double r251790 = 4.0;
        double r251791 = x;
        double r251792 = y;
        double r251793 = 0.75;
        double r251794 = r251792 * r251793;
        double r251795 = r251791 + r251794;
        double r251796 = z;
        double r251797 = r251795 - r251796;
        double r251798 = r251790 * r251797;
        double r251799 = r251798 / r251792;
        double r251800 = r251789 + r251799;
        return r251800;
}
↓
double f(double x, double y, double z) {
        double r251801 = 4.0;
        double r251802 = y;
        double r251803 = r251801 / r251802;
        double r251804 = x;
        double r251805 = z;
        double r251806 = r251804 - r251805;
        double r251807 = fma(r251803, r251806, r251801);
        return r251807;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.3

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{y}, \mathsf{fma}\left(0.75, y, x - z\right), 1\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 4\right) - 4 \cdot \frac{z}{y}}\]

  4. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{y}, x - z, 4\right)}\]

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))