Average Error: 0.2 → 0.0
Time: 9.0s
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 r222989 = 1.0;
        double r222990 = 4.0;
        double r222991 = x;
        double r222992 = y;
        double r222993 = 0.75;
        double r222994 = r222992 * r222993;
        double r222995 = r222991 + r222994;
        double r222996 = z;
        double r222997 = r222995 - r222996;
        double r222998 = r222990 * r222997;
        double r222999 = r222998 / r222992;
        double r223000 = r222989 + r222999;
        return r223000;
}
↓
double f(double x, double y, double z) {
        double r223001 = 4.0;
        double r223002 = x;
        double r223003 = y;
        double r223004 = r223002 / r223003;
        double r223005 = z;
        double r223006 = r223005 / r223003;
        double r223007 = r223004 - r223006;
        double r223008 = fma(r223001, r223007, r223001);
        return r223008;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)}\]
  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 2020046 +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)))