Average Error: 12.2 → 1.1
Time: 2.6s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\left(x \cdot \frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y + z}}{\sqrt[3]{z}}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\left(x \cdot \frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y + z}}{\sqrt[3]{z}}
double f(double x, double y, double z) {
        double r444697 = x;
        double r444698 = y;
        double r444699 = z;
        double r444700 = r444698 + r444699;
        double r444701 = r444697 * r444700;
        double r444702 = r444701 / r444699;
        return r444702;
}
↓
double f(double x, double y, double z) {
        double r444703 = x;
        double r444704 = y;
        double r444705 = z;
        double r444706 = r444704 + r444705;
        double r444707 = cbrt(r444706);
        double r444708 = r444707 * r444707;
        double r444709 = cbrt(r444705);
        double r444710 = r444709 * r444709;
        double r444711 = r444708 / r444710;
        double r444712 = r444703 * r444711;
        double r444713 = r444707 / r444709;
        double r444714 = r444712 * r444713;
        return r444714;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original12.2
Target3.2
Herbie1.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.2

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

  3. Applied *-un-lft-identity12.2

    \[\leadsto \frac{x \cdot \left(y + z\right)}{\color{blue}{1 \cdot z}}\]

  4. Applied times-frac3.5

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

  5. Simplified3.5

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

  7. Applied add-cube-cbrt4.7

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

  8. Applied add-cube-cbrt4.0

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

  9. Applied times-frac4.0

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

  10. Applied associate-*r*1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2019356 
(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))