Average Error: 12.1 → 1.0
Time: 3.4s
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 r367976 = x;
        double r367977 = y;
        double r367978 = z;
        double r367979 = r367977 + r367978;
        double r367980 = r367976 * r367979;
        double r367981 = r367980 / r367978;
        return r367981;
}
↓
double f(double x, double y, double z) {
        double r367982 = y;
        double r367983 = z;
        double r367984 = r367982 + r367983;
        double r367985 = cbrt(r367984);
        double r367986 = r367985 * r367985;
        double r367987 = cbrt(r367983);
        double r367988 = r367987 * r367987;
        double r367989 = r367986 / r367988;
        double r367990 = x;
        double r367991 = r367987 / r367985;
        double r367992 = r367990 / r367991;
        double r367993 = r367989 * r367992;
        return r367993;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.1
Target2.9
Herbie1.0
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.1

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

  3. Applied associate-/l*2.9

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

  5. Applied add-cube-cbrt4.1

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

  6. Applied add-cube-cbrt3.4

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

  7. Applied times-frac3.4

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

  8. Applied *-un-lft-identity3.4

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

  9. Applied times-frac1.0

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

  10. Simplified1.0

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

  11. Final simplification1.0

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

Reproduce

herbie shell --seed 2020024 
(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))