Average Error: 0.1 → 0.0
Time: 14.3s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)
double f(double x, double y, double z) {
        double r134930 = 1.0;
        double r134931 = 4.0;
        double r134932 = x;
        double r134933 = y;
        double r134934 = 0.25;
        double r134935 = r134933 * r134934;
        double r134936 = r134932 + r134935;
        double r134937 = z;
        double r134938 = r134936 - r134937;
        double r134939 = r134931 * r134938;
        double r134940 = r134939 / r134933;
        double r134941 = r134930 + r134940;
        return r134941;
}
↓
double f(double x, double y, double z) {
        double r134942 = 4.0;
        double r134943 = x;
        double r134944 = z;
        double r134945 = r134943 - r134944;
        double r134946 = y;
        double r134947 = r134945 / r134946;
        double r134948 = 2.0;
        double r134949 = fma(r134942, r134947, r134948);
        return r134949;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019322 +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)))