Average Error: 12.7 → 1.0
Time: 12.9s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\frac{\frac{x}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{y - z}}}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\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}}}
double f(double x, double y, double z) {
        double r598177 = x;
        double r598178 = y;
        double r598179 = z;
        double r598180 = r598178 - r598179;
        double r598181 = r598177 * r598180;
        double r598182 = r598181 / r598178;
        return r598182;
}
↓
double f(double x, double y, double z) {
        double r598183 = x;
        double r598184 = y;
        double r598185 = cbrt(r598184);
        double r598186 = r598185 * r598185;
        double r598187 = z;
        double r598188 = r598184 - r598187;
        double r598189 = cbrt(r598188);
        double r598190 = r598189 * r598189;
        double r598191 = r598186 / r598190;
        double r598192 = r598183 / r598191;
        double r598193 = r598185 / r598189;
        double r598194 = r598192 / r598193;
        return r598194;
}

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

Original12.7
Target3.3
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 12.7

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

  3. Applied associate-/l*3.2

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

  5. Applied add-cube-cbrt4.4

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

  6. Applied add-cube-cbrt3.7

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

  7. Applied times-frac3.7

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

  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. Final simplification1.0

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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))