Average Error: 0.2 → 0.0
Time: 20.3s
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 r205697 = 1.0;
        double r205698 = 4.0;
        double r205699 = x;
        double r205700 = y;
        double r205701 = 0.75;
        double r205702 = r205700 * r205701;
        double r205703 = r205699 + r205702;
        double r205704 = z;
        double r205705 = r205703 - r205704;
        double r205706 = r205698 * r205705;
        double r205707 = r205706 / r205700;
        double r205708 = r205697 + r205707;
        return r205708;
}
↓
double f(double x, double y, double z) {
        double r205709 = 4.0;
        double r205710 = x;
        double r205711 = y;
        double r205712 = r205710 / r205711;
        double r205713 = z;
        double r205714 = r205713 / r205711;
        double r205715 = r205712 - r205714;
        double r205716 = r205715 * r205709;
        double r205717 = r205709 + r205716;
        return r205717;
}

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)))