Average Error: 10.2 → 0.0
Time: 12.0s
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 r21050948 = x;
        double r21050949 = y;
        double r21050950 = z;
        double r21050951 = r21050950 - r21050948;
        double r21050952 = r21050949 * r21050951;
        double r21050953 = r21050948 + r21050952;
        double r21050954 = r21050953 / r21050950;
        return r21050954;
}
↓
double f(double x, double y, double z) {
        double r21050955 = x;
        double r21050956 = z;
        double r21050957 = r21050955 / r21050956;
        double r21050958 = y;
        double r21050959 = r21050957 + r21050958;
        double r21050960 = r21050957 * r21050958;
        double r21050961 = r21050959 - r21050960;
        return r21050961;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original10.2
Target0.0
Herbie0.0
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.2

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

  2. Taylor expanded around 0 3.5

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

  4. Applied div-inv3.5

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

  5. Taylor expanded around 0 3.5

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 
(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))