Average Error: 0.1 → 0.0
Time: 14.8s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 2\right) - \frac{4 \cdot z}{y}\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 2\right) - \frac{4 \cdot z}{y}
double f(double x, double y, double z) {
        double r198104 = 1.0;
        double r198105 = 4.0;
        double r198106 = x;
        double r198107 = y;
        double r198108 = 0.25;
        double r198109 = r198107 * r198108;
        double r198110 = r198106 + r198109;
        double r198111 = z;
        double r198112 = r198110 - r198111;
        double r198113 = r198105 * r198112;
        double r198114 = r198113 / r198107;
        double r198115 = r198104 + r198114;
        return r198115;
}
↓
double f(double x, double y, double z) {
        double r198116 = 4.0;
        double r198117 = x;
        double r198118 = y;
        double r198119 = r198117 / r198118;
        double r198120 = 2.0;
        double r198121 = fma(r198116, r198119, r198120);
        double r198122 = z;
        double r198123 = r198116 * r198122;
        double r198124 = r198123 / r198118;
        double r198125 = r198121 - r198124;
        return r198125;
}

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. Using strategy rm

  4. Applied div-sub0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))