Average Error: 12.3 → 1.1
Time: 3.3s
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 r421555 = x;
        double r421556 = y;
        double r421557 = z;
        double r421558 = r421556 + r421557;
        double r421559 = r421555 * r421558;
        double r421560 = r421559 / r421557;
        return r421560;
}
↓
double f(double x, double y, double z) {
        double r421561 = y;
        double r421562 = z;
        double r421563 = r421561 + r421562;
        double r421564 = cbrt(r421563);
        double r421565 = r421564 * r421564;
        double r421566 = cbrt(r421562);
        double r421567 = r421566 * r421566;
        double r421568 = r421565 / r421567;
        double r421569 = x;
        double r421570 = r421566 / r421564;
        double r421571 = r421569 / r421570;
        double r421572 = r421568 * r421571;
        return r421572;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.3
Target2.8
Herbie1.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.3

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

  3. Applied associate-/l*2.8

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

  5. Applied add-cube-cbrt4.0

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

  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-frac1.1

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

  10. Simplified1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2020056 
(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))