Average Error: 0.0 → 0.0
Time: 22.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)
double f(double x, double y, double z) {
        double r127844 = 1.0;
        double r127845 = 4.0;
        double r127846 = x;
        double r127847 = y;
        double r127848 = 0.25;
        double r127849 = r127847 * r127848;
        double r127850 = r127846 + r127849;
        double r127851 = z;
        double r127852 = r127850 - r127851;
        double r127853 = r127845 * r127852;
        double r127854 = r127853 / r127847;
        double r127855 = r127844 + r127854;
        return r127855;
}
↓
double f(double x, double y, double z) {
        double r127856 = 4.0;
        double r127857 = x;
        double r127858 = y;
        double r127859 = r127857 / r127858;
        double r127860 = z;
        double r127861 = r127860 / r127858;
        double r127862 = r127859 - r127861;
        double r127863 = 2.0;
        double r127864 = fma(r127856, r127862, r127863);
        return r127864;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)}\]
  5. Using strategy rm

  6. Applied div-sub0.0

    \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{y} - \frac{z}{y}}, 2\right)\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))