Average Error: 0.2 → 0.0
Time: 13.4s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4\right) - 4 \cdot \frac{z}{y}\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4\right) - 4 \cdot \frac{z}{y}
double f(double x, double y, double z) {
        double r274637 = 1.0;
        double r274638 = 4.0;
        double r274639 = x;
        double r274640 = y;
        double r274641 = 0.75;
        double r274642 = r274640 * r274641;
        double r274643 = r274639 + r274642;
        double r274644 = z;
        double r274645 = r274643 - r274644;
        double r274646 = r274638 * r274645;
        double r274647 = r274646 / r274640;
        double r274648 = r274637 + r274647;
        return r274648;
}
↓
double f(double x, double y, double z) {
        double r274649 = 4.0;
        double r274650 = x;
        double r274651 = y;
        double r274652 = r274650 / r274651;
        double r274653 = fma(r274649, r274652, r274649);
        double r274654 = z;
        double r274655 = r274654 / r274651;
        double r274656 = r274649 * r274655;
        double r274657 = r274653 - r274656;
        return r274657;
}

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

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

  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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +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)))