Average Error: 10.5 → 0.0
Time: 9.0s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(y + \frac{x}{z}\right) - \frac{x}{z} \cdot y\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(y + \frac{x}{z}\right) - \frac{x}{z} \cdot y
double f(double x, double y, double z) {
        double r646237 = x;
        double r646238 = y;
        double r646239 = z;
        double r646240 = r646239 - r646237;
        double r646241 = r646238 * r646240;
        double r646242 = r646237 + r646241;
        double r646243 = r646242 / r646239;
        return r646243;
}
↓
double f(double x, double y, double z) {
        double r646244 = y;
        double r646245 = x;
        double r646246 = z;
        double r646247 = r646245 / r646246;
        double r646248 = r646244 + r646247;
        double r646249 = r646247 * r646244;
        double r646250 = r646248 - r646249;
        return r646250;
}

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. Simplified3.9

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

  4. Applied div-sub3.9

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

  5. Taylor expanded around 0 3.9

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

  6. Simplified0.0

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

  8. Applied div-inv0.1

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 
(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))