Average Error: 0.2 → 0.0
Time: 11.2s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 4\right)
double f(double x, double y, double z) {
        double r256583 = 1.0;
        double r256584 = 4.0;
        double r256585 = x;
        double r256586 = y;
        double r256587 = 0.75;
        double r256588 = r256586 * r256587;
        double r256589 = r256585 + r256588;
        double r256590 = z;
        double r256591 = r256589 - r256590;
        double r256592 = r256584 * r256591;
        double r256593 = r256592 / r256586;
        double r256594 = r256583 + r256593;
        return r256594;
}
↓
double f(double x, double y, double z) {
        double r256595 = x;
        double r256596 = y;
        double r256597 = r256595 / r256596;
        double r256598 = z;
        double r256599 = r256598 / r256596;
        double r256600 = r256597 - r256599;
        double r256601 = 4.0;
        double r256602 = fma(r256600, r256601, r256601);
        return r256602;
}

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

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

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))