Average Error: 10.3 → 0.0
Time: 32.2s
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 r38668115 = x;
        double r38668116 = y;
        double r38668117 = z;
        double r38668118 = r38668117 - r38668115;
        double r38668119 = r38668116 * r38668118;
        double r38668120 = r38668115 + r38668119;
        double r38668121 = r38668120 / r38668117;
        return r38668121;
}
↓
double f(double x, double y, double z) {
        double r38668122 = x;
        double r38668123 = z;
        double r38668124 = r38668122 / r38668123;
        double r38668125 = y;
        double r38668126 = r38668124 + r38668125;
        double r38668127 = r38668124 * r38668125;
        double r38668128 = r38668126 - r38668127;
        return r38668128;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original10.3
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.3

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

  2. Taylor expanded around 0 3.3

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

  4. Applied add-cube-cbrt3.5

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

  5. Applied times-frac0.9

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

  6. Taylor expanded around 0 3.3

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

  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 2019168 
(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))