Average Error: 7.9 → 6.7
Time: 12.9s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\sqrt[3]{{\left(\mathsf{fma}\left(\frac{{x0}^{\frac{2}{3}}}{\sqrt{1} + \sqrt{x1}}, \frac{1}{\frac{\sqrt{1} - \sqrt{x1}}{\sqrt[3]{x0}}}, -x0\right)\right)}^{3}}\]
\frac{x0}{1 - x1} - x0
↓
\sqrt[3]{{\left(\mathsf{fma}\left(\frac{{x0}^{\frac{2}{3}}}{\sqrt{1} + \sqrt{x1}}, \frac{1}{\frac{\sqrt{1} - \sqrt{x1}}{\sqrt[3]{x0}}}, -x0\right)\right)}^{3}}
double f(double x0, double x1) {
        double r148811 = x0;
        double r148812 = 1.0;
        double r148813 = x1;
        double r148814 = r148812 - r148813;
        double r148815 = r148811 / r148814;
        double r148816 = r148815 - r148811;
        return r148816;
}
↓
double f(double x0, double x1) {
        double r148817 = x0;
        double r148818 = 0.6666666666666666;
        double r148819 = pow(r148817, r148818);
        double r148820 = 1.0;
        double r148821 = sqrt(r148820);
        double r148822 = x1;
        double r148823 = sqrt(r148822);
        double r148824 = r148821 + r148823;
        double r148825 = r148819 / r148824;
        double r148826 = 1.0;
        double r148827 = r148821 - r148823;
        double r148828 = cbrt(r148817);
        double r148829 = r148827 / r148828;
        double r148830 = r148826 / r148829;
        double r148831 = -r148817;
        double r148832 = fma(r148825, r148830, r148831);
        double r148833 = 3.0;
        double r148834 = pow(r148832, r148833);
        double r148835 = cbrt(r148834);
        return r148835;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
Target0.3
Herbie6.7
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

    \[\frac{x0}{1 - x1} - x0\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt7.9

    \[\leadsto \frac{x0}{1 - \color{blue}{\sqrt{x1} \cdot \sqrt{x1}}} - x0\]

  4. Applied add-sqr-sqrt7.9

    \[\leadsto \frac{x0}{\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \sqrt{x1} \cdot \sqrt{x1}} - x0\]

  5. Applied difference-of-squares7.9

    \[\leadsto \frac{x0}{\color{blue}{\left(\sqrt{1} + \sqrt{x1}\right) \cdot \left(\sqrt{1} - \sqrt{x1}\right)}} - x0\]

  6. Applied add-cube-cbrt7.9

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x0} \cdot \sqrt[3]{x0}\right) \cdot \sqrt[3]{x0}}}{\left(\sqrt{1} + \sqrt{x1}\right) \cdot \left(\sqrt{1} - \sqrt{x1}\right)} - x0\]

  7. Applied times-frac7.8

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{1} + \sqrt{x1}} \cdot \frac{\sqrt[3]{x0}}{\sqrt{1} - \sqrt{x1}}} - x0\]

  8. Applied fma-neg7.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt[3]{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)}\]
  9. Using strategy rm

  10. Applied add-cbrt-cube7.4

    \[\leadsto \color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt[3]{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right) \cdot \mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt[3]{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\right) \cdot \mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt[3]{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)}}\]

  11. Simplified7.4

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(\frac{{x0}^{\frac{2}{3}}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt[3]{x0}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\right)}^{3}}}\]
  12. Using strategy rm

  13. Applied *-un-lft-identity7.4

    \[\leadsto \sqrt[3]{{\left(\mathsf{fma}\left(\frac{{x0}^{\frac{2}{3}}}{\sqrt{1} + \sqrt{x1}}, \frac{\sqrt[3]{\color{blue}{1 \cdot x0}}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\right)}^{3}}\]

  14. Applied cbrt-prod7.4

    \[\leadsto \sqrt[3]{{\left(\mathsf{fma}\left(\frac{{x0}^{\frac{2}{3}}}{\sqrt{1} + \sqrt{x1}}, \frac{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{x0}}}{\sqrt{1} - \sqrt{x1}}, -x0\right)\right)}^{3}}\]

  15. Applied associate-/l*6.7

    \[\leadsto \sqrt[3]{{\left(\mathsf{fma}\left(\frac{{x0}^{\frac{2}{3}}}{\sqrt{1} + \sqrt{x1}}, \color{blue}{\frac{\sqrt[3]{1}}{\frac{\sqrt{1} - \sqrt{x1}}{\sqrt[3]{x0}}}}, -x0\right)\right)}^{3}}\]

  16. Final simplification6.7

    \[\leadsto \sqrt[3]{{\left(\mathsf{fma}\left(\frac{{x0}^{\frac{2}{3}}}{\sqrt{1} + \sqrt{x1}}, \frac{1}{\frac{\sqrt{1} - \sqrt{x1}}{\sqrt[3]{x0}}}, -x0\right)\right)}^{3}}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :pre (or (and (== x0 1.855) (== x1 0.000209)) (and (== x0 2.985) (== x1 0.0186)))

  :herbie-target
  (/ (* x0 x1) (- 1 x1))

  (- (/ x0 (- 1 x1)) x0))