Average Error: 10.5 → 0.0
Time: 38.6s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) - \frac{x}{z} \cdot y\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) - \frac{x}{z} \cdot y
double f(double x, double y, double z) {
        double r35670287 = x;
        double r35670288 = y;
        double r35670289 = z;
        double r35670290 = r35670289 - r35670287;
        double r35670291 = r35670288 * r35670290;
        double r35670292 = r35670287 + r35670291;
        double r35670293 = r35670292 / r35670289;
        return r35670293;
}
↓
double f(double x, double y, double z) {
        double r35670294 = x;
        double r35670295 = z;
        double r35670296 = r35670294 / r35670295;
        double r35670297 = y;
        double r35670298 = r35670296 + r35670297;
        double r35670299 = r35670296 * r35670297;
        double r35670300 = r35670298 - r35670299;
        return r35670300;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original10.5
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.5

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

  2. Taylor expanded around 0 3.6

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

  4. Applied add-cube-cbrt3.7

    \[\leadsto \left(y + \frac{x}{z}\right) - \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

  5. Applied times-frac1.0

    \[\leadsto \left(y + \frac{x}{z}\right) - \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}}\]

  6. Taylor expanded around 0 3.6

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))