Average Error: 0.1 → 0.1
Time: 14.5s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[2 + \frac{\left(x - z\right) \cdot 4}{y}\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
2 + \frac{\left(x - z\right) \cdot 4}{y}
double f(double x, double y, double z) {
        double r11959888 = 1.0;
        double r11959889 = 4.0;
        double r11959890 = x;
        double r11959891 = y;
        double r11959892 = 0.25;
        double r11959893 = r11959891 * r11959892;
        double r11959894 = r11959890 + r11959893;
        double r11959895 = z;
        double r11959896 = r11959894 - r11959895;
        double r11959897 = r11959889 * r11959896;
        double r11959898 = r11959897 / r11959891;
        double r11959899 = r11959888 + r11959898;
        return r11959899;
}
↓
double f(double x, double y, double z) {
        double r11959900 = 2.0;
        double r11959901 = x;
        double r11959902 = z;
        double r11959903 = r11959901 - r11959902;
        double r11959904 = 4.0;
        double r11959905 = r11959903 * r11959904;
        double r11959906 = y;
        double r11959907 = r11959905 / r11959906;
        double r11959908 = r11959900 + r11959907;
        return r11959908;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

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. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019169 +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)))