Average Error: 0.2 → 0.0
Time: 21.3s
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 r200763 = 1.0;
        double r200764 = 4.0;
        double r200765 = x;
        double r200766 = y;
        double r200767 = 0.75;
        double r200768 = r200766 * r200767;
        double r200769 = r200765 + r200768;
        double r200770 = z;
        double r200771 = r200769 - r200770;
        double r200772 = r200764 * r200771;
        double r200773 = r200772 / r200766;
        double r200774 = r200763 + r200773;
        return r200774;
}
↓
double f(double x, double y, double z) {
        double r200775 = 4.0;
        double r200776 = x;
        double r200777 = y;
        double r200778 = r200776 / r200777;
        double r200779 = z;
        double r200780 = r200779 / r200777;
        double r200781 = r200778 - r200780;
        double r200782 = fma(r200775, r200781, r200775);
        return r200782;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75 + \frac{x - z}{y}, 4, 1\right)}\]

  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 2019326 +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)))