Average Error: 0.2 → 0.0
Time: 12.7s
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 r11824479 = 1.0;
        double r11824480 = 4.0;
        double r11824481 = x;
        double r11824482 = y;
        double r11824483 = 0.75;
        double r11824484 = r11824482 * r11824483;
        double r11824485 = r11824481 + r11824484;
        double r11824486 = z;
        double r11824487 = r11824485 - r11824486;
        double r11824488 = r11824480 * r11824487;
        double r11824489 = r11824488 / r11824482;
        double r11824490 = r11824479 + r11824489;
        return r11824490;
}
↓
double f(double x, double y, double z) {
        double r11824491 = 4.0;
        double r11824492 = x;
        double r11824493 = y;
        double r11824494 = r11824492 / r11824493;
        double r11824495 = z;
        double r11824496 = r11824495 / r11824493;
        double r11824497 = r11824494 - r11824496;
        double r11824498 = fma(r11824491, r11824497, r11824491);
        return r11824498;
}

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

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

  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

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

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +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)))