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 r235460 = 1.0;
        double r235461 = 4.0;
        double r235462 = x;
        double r235463 = y;
        double r235464 = 0.75;
        double r235465 = r235463 * r235464;
        double r235466 = r235462 + r235465;
        double r235467 = z;
        double r235468 = r235466 - r235467;
        double r235469 = r235461 * r235468;
        double r235470 = r235469 / r235463;
        double r235471 = r235460 + r235470;
        return r235471;
}
↓
double f(double x, double y, double z) {
        double r235472 = 4.0;
        double r235473 = x;
        double r235474 = y;
        double r235475 = r235473 / r235474;
        double r235476 = z;
        double r235477 = r235476 / r235474;
        double r235478 = r235475 - r235477;
        double r235479 = fma(r235472, r235478, r235472);
        return r235479;
}

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 2019196 +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)))