Average Error: 10.4 → 0.2
Time: 12.1s
Precision: 64
\[\frac{x + y \cdot \left(z - x\right)}{z}\]
↓
\[\left(\frac{x}{z} + y\right) - \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot y\right) \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)\]
\frac{x + y \cdot \left(z - x\right)}{z}
↓
\left(\frac{x}{z} + y\right) - \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot y\right) \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right)
double f(double x, double y, double z) {
        double r37282384 = x;
        double r37282385 = y;
        double r37282386 = z;
        double r37282387 = r37282386 - r37282384;
        double r37282388 = r37282385 * r37282387;
        double r37282389 = r37282384 + r37282388;
        double r37282390 = r37282389 / r37282386;
        return r37282390;
}
↓
double f(double x, double y, double z) {
        double r37282391 = x;
        double r37282392 = z;
        double r37282393 = r37282391 / r37282392;
        double r37282394 = y;
        double r37282395 = r37282393 + r37282394;
        double r37282396 = cbrt(r37282391);
        double r37282397 = cbrt(r37282392);
        double r37282398 = r37282396 / r37282397;
        double r37282399 = r37282398 * r37282394;
        double r37282400 = r37282398 * r37282398;
        double r37282401 = r37282399 * r37282400;
        double r37282402 = r37282395 - r37282401;
        return r37282402;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

Enter valid numbers for all inputs

Target

Original10.4
Target0.0
Herbie0.2
\[\left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}}\]

Derivation

  1. Initial program 10.4

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

  2. Taylor expanded around 0 3.4

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

  4. Applied associate-/l*2.9

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

  6. Applied *-un-lft-identity2.9

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

  7. Applied add-cube-cbrt3.0

    \[\leadsto \left(y + \frac{x}{z}\right) - \frac{x}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{1 \cdot y}}\]

  8. Applied times-frac3.0

    \[\leadsto \left(y + \frac{x}{z}\right) - \frac{x}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1} \cdot \frac{\sqrt[3]{z}}{y}}}\]

  9. Applied add-cube-cbrt3.1

    \[\leadsto \left(y + \frac{x}{z}\right) - \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1} \cdot \frac{\sqrt[3]{z}}{y}}\]

  10. Applied times-frac0.7

    \[\leadsto \left(y + \frac{x}{z}\right) - \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{1}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{y}}}\]

  11. Simplified0.7

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019192 
(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))