Average Error: 12.5 → 1.0
Time: 2.5s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\left(x \cdot \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{y}}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\left(x \cdot \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{y}}
double f(double x, double y, double z) {
        double r825132 = x;
        double r825133 = y;
        double r825134 = z;
        double r825135 = r825133 - r825134;
        double r825136 = r825132 * r825135;
        double r825137 = r825136 / r825133;
        return r825137;
}
↓
double f(double x, double y, double z) {
        double r825138 = x;
        double r825139 = y;
        double r825140 = z;
        double r825141 = r825139 - r825140;
        double r825142 = cbrt(r825141);
        double r825143 = r825142 * r825142;
        double r825144 = cbrt(r825139);
        double r825145 = r825144 * r825144;
        double r825146 = r825143 / r825145;
        double r825147 = r825138 * r825146;
        double r825148 = r825142 / r825144;
        double r825149 = r825147 * r825148;
        return r825149;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.5
Target3.1
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Initial program 12.5

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

  3. Applied *-un-lft-identity12.5

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

  4. Applied times-frac3.5

    \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y - z}{y}}\]

  5. Simplified3.5

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

  7. Applied add-cube-cbrt4.7

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

  8. Applied add-cube-cbrt4.0

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

  9. Applied times-frac4.0

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

  10. Applied associate-*r*1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2019356 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :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))