Average Error: 12.6 → 1.2
Time: 12.5s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y + z}}}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y + z}}}
double f(double x, double y, double z) {
        double r277874 = x;
        double r277875 = y;
        double r277876 = z;
        double r277877 = r277875 + r277876;
        double r277878 = r277874 * r277877;
        double r277879 = r277878 / r277876;
        return r277879;
}
↓
double f(double x, double y, double z) {
        double r277880 = y;
        double r277881 = z;
        double r277882 = r277880 + r277881;
        double r277883 = cbrt(r277882);
        double r277884 = r277883 * r277883;
        double r277885 = cbrt(r277881);
        double r277886 = r277885 * r277885;
        double r277887 = r277884 / r277886;
        double r277888 = x;
        double r277889 = r277885 / r277883;
        double r277890 = r277888 / r277889;
        double r277891 = r277887 * r277890;
        return r277891;
}

Error

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Bits error versus y

Bits error versus z

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Results

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Target

Original12.6
Target3.2
Herbie1.2
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.6

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

  3. Applied associate-/l*3.2

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

  5. Applied add-cube-cbrt4.4

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

  6. Applied add-cube-cbrt3.7

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

  7. Applied times-frac3.7

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

  8. Applied *-un-lft-identity3.7

    \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y + z}}}\]

  9. Applied times-frac1.2

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y + z}}}}\]

  10. Simplified1.2

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

  11. Final simplification1.2

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))