Average Error: 0.2 → 0.0
Time: 19.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[4 + \left(\frac{x}{y} - \frac{z}{y}\right) \cdot 4\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
4 + \left(\frac{x}{y} - \frac{z}{y}\right) \cdot 4
double f(double x, double y, double z) {
        double r225832 = 1.0;
        double r225833 = 4.0;
        double r225834 = x;
        double r225835 = y;
        double r225836 = 0.75;
        double r225837 = r225835 * r225836;
        double r225838 = r225834 + r225837;
        double r225839 = z;
        double r225840 = r225838 - r225839;
        double r225841 = r225833 * r225840;
        double r225842 = r225841 / r225835;
        double r225843 = r225832 + r225842;
        return r225843;
}
↓
double f(double x, double y, double z) {
        double r225844 = 4.0;
        double r225845 = x;
        double r225846 = y;
        double r225847 = r225845 / r225846;
        double r225848 = z;
        double r225849 = r225848 / r225846;
        double r225850 = r225847 - r225849;
        double r225851 = r225850 * r225844;
        double r225852 = r225844 + r225851;
        return r225852;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{1 + \left(\frac{x - z}{y} + 0.75\right) \cdot 4}\]
  3. Using strategy rm

  4. Applied div-sub0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 
(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)))