Average Error: 0.1 → 0.0
Time: 14.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[2 + 4 \cdot \left(\frac{x}{y} - \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
2 + 4 \cdot \left(\frac{x}{y} - \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r38067272 = 1.0;
        double r38067273 = 4.0;
        double r38067274 = x;
        double r38067275 = y;
        double r38067276 = 0.25;
        double r38067277 = r38067275 * r38067276;
        double r38067278 = r38067274 + r38067277;
        double r38067279 = z;
        double r38067280 = r38067278 - r38067279;
        double r38067281 = r38067273 * r38067280;
        double r38067282 = r38067281 / r38067275;
        double r38067283 = r38067272 + r38067282;
        return r38067283;
}
↓
double f(double x, double y, double z) {
        double r38067284 = 2.0;
        double r38067285 = 4.0;
        double r38067286 = x;
        double r38067287 = y;
        double r38067288 = r38067286 / r38067287;
        double r38067289 = z;
        double r38067290 = r38067289 / r38067287;
        double r38067291 = r38067288 - r38067290;
        double r38067292 = r38067285 * r38067291;
        double r38067293 = r38067284 + r38067292;
        return r38067293;
}

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

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

  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

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

Reproduce

herbie shell --seed 2019173 
(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)))