Average Error: 12.9 → 1.4
Time: 47.4s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\left(\left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right) \cdot y\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}} + x\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\left(\left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right) \cdot y\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}} + x
double f(double x, double y, double z) {
        double r21048632 = x;
        double r21048633 = y;
        double r21048634 = z;
        double r21048635 = r21048633 + r21048634;
        double r21048636 = r21048632 * r21048635;
        double r21048637 = r21048636 / r21048634;
        return r21048637;
}
↓
double f(double x, double y, double z) {
        double r21048638 = x;
        double r21048639 = cbrt(r21048638);
        double r21048640 = z;
        double r21048641 = cbrt(r21048640);
        double r21048642 = r21048639 / r21048641;
        double r21048643 = r21048642 * r21048642;
        double r21048644 = y;
        double r21048645 = r21048643 * r21048644;
        double r21048646 = r21048645 * r21048642;
        double r21048647 = r21048646 + r21048638;
        return r21048647;
}

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
Target2.8
Herbie1.4
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.9

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

  2. Simplified5.0

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

  4. Applied fma-udef5.0

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

  6. Applied add-cube-cbrt5.4

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

  7. Applied add-cube-cbrt5.4

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

  8. Applied times-frac5.4

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

  9. Applied associate-*r*1.4

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

  10. Simplified1.4

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

  11. Final simplification1.4

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))