Average Error: 0.1 → 0.0
Time: 50.6s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[2 + \left(\frac{x}{y} - \frac{z}{y}\right) \cdot 4\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
2 + \left(\frac{x}{y} - \frac{z}{y}\right) \cdot 4
double f(double x, double y, double z) {
        double r14895983 = 1.0;
        double r14895984 = 4.0;
        double r14895985 = x;
        double r14895986 = y;
        double r14895987 = 0.25;
        double r14895988 = r14895986 * r14895987;
        double r14895989 = r14895985 + r14895988;
        double r14895990 = z;
        double r14895991 = r14895989 - r14895990;
        double r14895992 = r14895984 * r14895991;
        double r14895993 = r14895992 / r14895986;
        double r14895994 = r14895983 + r14895993;
        return r14895994;
}
↓
double f(double x, double y, double z) {
        double r14895995 = 2.0;
        double r14895996 = x;
        double r14895997 = y;
        double r14895998 = r14895996 / r14895997;
        double r14895999 = z;
        double r14896000 = r14895999 / r14895997;
        double r14896001 = r14895998 - r14896000;
        double r14896002 = 4.0;
        double r14896003 = r14896001 * r14896002;
        double r14896004 = r14895995 + r14896003;
        return r14896004;
}

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 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

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

  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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 
(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)))