Average Error: 12.9 → 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 r469135 = x;
        double r469136 = y;
        double r469137 = z;
        double r469138 = r469136 + r469137;
        double r469139 = r469135 * r469138;
        double r469140 = r469139 / r469137;
        return r469140;
}
↓
double f(double x, double y, double z) {
        double r469141 = x;
        double r469142 = y;
        double r469143 = z;
        double r469144 = r469142 + r469143;
        double r469145 = cbrt(r469144);
        double r469146 = r469145 * r469145;
        double r469147 = cbrt(r469143);
        double r469148 = r469147 * r469147;
        double r469149 = r469146 / r469148;
        double r469150 = r469141 * r469149;
        double r469151 = r469145 / r469147;
        double r469152 = r469150 * r469151;
        return r469152;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

Enter valid numbers for all inputs

Target

Original12.9
Target3.0
Herbie1.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.9

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

  3. Applied *-un-lft-identity12.9

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

  4. Applied times-frac3.2

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

  5. Simplified3.2

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

  7. Applied add-cube-cbrt4.4

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

  8. Applied add-cube-cbrt3.7

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

  9. Applied times-frac3.7

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

  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 2020039 
(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))