Average Error: 0.2 → 0.0
Time: 15.4s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)\]
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 r11810537 = 1.0;
        double r11810538 = 4.0;
        double r11810539 = x;
        double r11810540 = y;
        double r11810541 = 0.75;
        double r11810542 = r11810540 * r11810541;
        double r11810543 = r11810539 + r11810542;
        double r11810544 = z;
        double r11810545 = r11810543 - r11810544;
        double r11810546 = r11810538 * r11810545;
        double r11810547 = r11810546 / r11810540;
        double r11810548 = r11810537 + r11810547;
        return r11810548;
}
↓
double f(double x, double y, double z) {
        double r11810549 = 4.0;
        double r11810550 = x;
        double r11810551 = y;
        double r11810552 = r11810550 / r11810551;
        double r11810553 = z;
        double r11810554 = r11810553 / r11810551;
        double r11810555 = r11810552 - r11810554;
        double r11810556 = fma(r11810549, r11810555, r11810549);
        return r11810556;
}

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

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

  5. Taylor expanded around 0 0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +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)))