Average Error: 0.1 → 0.2
Time: 1.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{4}{y}, \mathsf{fma}\left(0.25, y, x - z\right), 1\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{4}{y}, \mathsf{fma}\left(0.25, y, x - z\right), 1\right)
double f(double x, double y, double z) {
        double r340784 = 1.0;
        double r340785 = 4.0;
        double r340786 = x;
        double r340787 = y;
        double r340788 = 0.25;
        double r340789 = r340787 * r340788;
        double r340790 = r340786 + r340789;
        double r340791 = z;
        double r340792 = r340790 - r340791;
        double r340793 = r340785 * r340792;
        double r340794 = r340793 / r340787;
        double r340795 = r340784 + r340794;
        return r340795;
}
↓
double f(double x, double y, double z) {
        double r340796 = 4.0;
        double r340797 = y;
        double r340798 = r340796 / r340797;
        double r340799 = 0.25;
        double r340800 = x;
        double r340801 = z;
        double r340802 = r340800 - r340801;
        double r340803 = fma(r340799, r340797, r340802);
        double r340804 = 1.0;
        double r340805 = fma(r340798, r340803, r340804);
        return r340805;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{y}, \mathsf{fma}\left(0.25, y, x - z\right), 1\right)}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{4}{y}, \mathsf{fma}\left(0.25, y, x - z\right), 1\right)\]

Reproduce

herbie shell --seed 2020081 +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)))