Average Error: 0.1 → 0.0
Time: 10.5s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)
double f(double x, double y, double z) {
        double r373763 = 1.0;
        double r373764 = 4.0;
        double r373765 = x;
        double r373766 = y;
        double r373767 = 0.25;
        double r373768 = r373766 * r373767;
        double r373769 = r373765 + r373768;
        double r373770 = z;
        double r373771 = r373769 - r373770;
        double r373772 = r373764 * r373771;
        double r373773 = r373772 / r373766;
        double r373774 = r373763 + r373773;
        return r373774;
}
↓
double f(double x, double y, double z) {
        double r373775 = x;
        double r373776 = z;
        double r373777 = r373775 - r373776;
        double r373778 = y;
        double r373779 = r373777 / r373778;
        double r373780 = 4.0;
        double r373781 = 2.0;
        double r373782 = fma(r373779, r373780, r373781);
        return r373782;
}

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

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

  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

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

  5. Final simplification0.0

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

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, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))