Average Error: 0.2 → 0.0
Time: 14.2s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)
double f(double x, double y, double z) {
        double r15274646 = 1.0;
        double r15274647 = 4.0;
        double r15274648 = x;
        double r15274649 = y;
        double r15274650 = 0.75;
        double r15274651 = r15274649 * r15274650;
        double r15274652 = r15274648 + r15274651;
        double r15274653 = z;
        double r15274654 = r15274652 - r15274653;
        double r15274655 = r15274647 * r15274654;
        double r15274656 = r15274655 / r15274649;
        double r15274657 = r15274646 + r15274656;
        return r15274657;
}
↓
double f(double x, double y, double z) {
        double r15274658 = 4.0;
        double r15274659 = x;
        double r15274660 = y;
        double r15274661 = r15274659 / r15274660;
        double r15274662 = z;
        double r15274663 = r15274662 / r15274660;
        double r15274664 = r15274661 - r15274663;
        double r15274665 = fma(r15274658, r15274664, r15274658);
        return r15274665;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

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

  2. Simplified0.0

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))