Average Error: 12.3 → 0.9
Time: 15.1s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{\frac{x}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}} + x\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{\frac{x}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}} + x
double f(double x, double y, double z) {
        double r248734 = x;
        double r248735 = y;
        double r248736 = z;
        double r248737 = r248735 + r248736;
        double r248738 = r248734 * r248737;
        double r248739 = r248738 / r248736;
        return r248739;
}
↓
double f(double x, double y, double z) {
        double r248740 = x;
        double r248741 = z;
        double r248742 = cbrt(r248741);
        double r248743 = r248742 * r248742;
        double r248744 = y;
        double r248745 = cbrt(r248744);
        double r248746 = r248745 * r248745;
        double r248747 = r248743 / r248746;
        double r248748 = r248740 / r248747;
        double r248749 = r248742 / r248745;
        double r248750 = r248748 / r248749;
        double r248751 = r248750 + r248740;
        return r248751;
}

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
Target3.3
Herbie0.9
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.3

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

  2. Simplified3.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)}\]

  3. Taylor expanded around 0 4.9

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

  4. Simplified4.7

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

  6. Applied fma-udef4.7

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

  7. Simplified4.9

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

  9. Applied associate-/l*3.3

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

  11. Applied add-cube-cbrt3.6

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

  12. Applied add-cube-cbrt3.7

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

  13. Applied times-frac3.7

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

  14. Applied associate-/r*0.9

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

  15. Final simplification0.9

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

Reproduce

herbie shell --seed 2019347 +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))