Average Error: 12.9 → 1.1
Time: 3.2s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot x\right) + x\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot x\right) + x
double f(double x, double y, double z) {
        double r377870 = x;
        double r377871 = y;
        double r377872 = z;
        double r377873 = r377871 + r377872;
        double r377874 = r377870 * r377873;
        double r377875 = r377874 / r377872;
        return r377875;
}
↓
double f(double x, double y, double z) {
        double r377876 = y;
        double r377877 = cbrt(r377876);
        double r377878 = r377877 * r377877;
        double r377879 = z;
        double r377880 = cbrt(r377879);
        double r377881 = r377880 * r377880;
        double r377882 = r377878 / r377881;
        double r377883 = r377877 / r377880;
        double r377884 = x;
        double r377885 = r377883 * r377884;
        double r377886 = r377882 * r377885;
        double r377887 = r377886 + r377884;
        return r377887;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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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. Simplified3.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)}\]
  3. Using strategy rm

  4. Applied fma-udef3.2

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

  6. Applied add-cube-cbrt3.6

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

  7. Applied add-cube-cbrt3.7

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

  8. Applied times-frac3.7

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

  9. Applied associate-*l*1.1

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

  10. Final simplification1.1

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(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))