Average Error: 12.2 → 0.7
Time: 15.0s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\frac{\frac{\frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\frac{\frac{\frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}
double f(double x, double y, double z) {
        double r39452471 = x;
        double r39452472 = y;
        double r39452473 = z;
        double r39452474 = r39452472 - r39452473;
        double r39452475 = r39452471 * r39452474;
        double r39452476 = r39452475 / r39452472;
        return r39452476;
}
↓
double f(double x, double y, double z) {
        double r39452477 = x;
        double r39452478 = y;
        double r39452479 = cbrt(r39452478);
        double r39452480 = z;
        double r39452481 = r39452478 - r39452480;
        double r39452482 = cbrt(r39452481);
        double r39452483 = r39452479 / r39452482;
        double r39452484 = r39452477 / r39452483;
        double r39452485 = r39452484 / r39452483;
        double r39452486 = r39452485 / r39452483;
        return r39452486;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.2
Target2.9
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Initial program 12.2

    \[\frac{x \cdot \left(y - z\right)}{y}\]
  2. Using strategy rm

  3. Applied associate-/l*2.9

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

  5. Applied add-cube-cbrt4.1

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

  6. Applied add-cube-cbrt3.4

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

  7. Applied times-frac3.4

    \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}\]

  8. Applied associate-/r*1.0

    \[\leadsto \color{blue}{\frac{\frac{x}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}\]

  9. Simplified0.7

    \[\leadsto \frac{\color{blue}{\frac{\frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}\]

  10. Final simplification0.7

    \[\leadsto \frac{\frac{\frac{x}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

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