Average Error: 12.4 → 1.2
Time: 3.5s
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 r432218 = x;
        double r432219 = y;
        double r432220 = z;
        double r432221 = r432219 + r432220;
        double r432222 = r432218 * r432221;
        double r432223 = r432222 / r432220;
        return r432223;
}
↓
double f(double x, double y, double z) {
        double r432224 = y;
        double r432225 = z;
        double r432226 = r432224 + r432225;
        double r432227 = cbrt(r432226);
        double r432228 = r432227 * r432227;
        double r432229 = cbrt(r432225);
        double r432230 = r432229 * r432229;
        double r432231 = r432228 / r432230;
        double r432232 = x;
        double r432233 = r432229 / r432227;
        double r432234 = r432232 / r432233;
        double r432235 = r432231 * r432234;
        return r432235;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.4
Target3.2
Herbie1.2
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.4

    \[\frac{x \cdot \left(y + z\right)}{z}\]
  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-frac3.8

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

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

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

  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 2019322 
(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))