Average Error: 7.9 → 6.4
Time: 3.8s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[e^{\log \left(\log \left(\frac{1}{\sqrt{e^{x0}}}\right) + \mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)\right)}\]
\frac{x0}{1 - x1} - x0
↓
e^{\log \left(\log \left(\frac{1}{\sqrt{e^{x0}}}\right) + \mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)\right)}
double f(double x0, double x1) {
        double r211636 = x0;
        double r211637 = 1.0;
        double r211638 = x1;
        double r211639 = r211637 - r211638;
        double r211640 = r211636 / r211639;
        double r211641 = r211640 - r211636;
        return r211641;
}
↓
double f(double x0, double x1) {
        double r211642 = 1.0;
        double r211643 = x0;
        double r211644 = exp(r211643);
        double r211645 = sqrt(r211644);
        double r211646 = r211642 / r211645;
        double r211647 = log(r211646);
        double r211648 = cbrt(r211643);
        double r211649 = 1.0;
        double r211650 = x1;
        double r211651 = r211649 - r211650;
        double r211652 = r211648 / r211651;
        double r211653 = 0.6666666666666666;
        double r211654 = pow(r211643, r211653);
        double r211655 = fma(r211652, r211654, r211647);
        double r211656 = r211647 + r211655;
        double r211657 = log(r211656);
        double r211658 = exp(r211657);
        return r211658;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.9
Target0.2
Herbie6.4
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

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

  3. Applied *-un-lft-identity7.9

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

  4. Applied add-cube-cbrt7.9

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

  5. Applied times-frac8.2

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

  6. Applied fma-neg6.9

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

  8. Applied add-exp-log6.9

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

  10. Applied add-log-exp7.8

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

  11. Simplified6.7

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

  13. Applied add-sqr-sqrt7.2

    \[\leadsto e^{\log \left(\log \left(\frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{\color{blue}{\sqrt{e^{x0}} \cdot \sqrt{e^{x0}}}}\right)\right)}\]

  14. Applied *-un-lft-identity7.2

    \[\leadsto e^{\log \left(\log \left(\frac{{\color{blue}{\left(1 \cdot e^{{x0}^{\frac{2}{3}}}\right)}}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{\sqrt{e^{x0}} \cdot \sqrt{e^{x0}}}\right)\right)}\]

  15. Applied unpow-prod-down7.2

    \[\leadsto e^{\log \left(\log \left(\frac{\color{blue}{{1}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)} \cdot {\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}}{\sqrt{e^{x0}} \cdot \sqrt{e^{x0}}}\right)\right)}\]

  16. Applied times-frac6.8

    \[\leadsto e^{\log \left(\log \color{blue}{\left(\frac{{1}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{\sqrt{e^{x0}}} \cdot \frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{\sqrt{e^{x0}}}\right)}\right)}\]

  17. Applied log-prod7.1

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

  18. Simplified7.1

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

  19. Simplified6.4

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

  20. Final simplification6.4

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

Reproduce

herbie shell --seed 2020057 +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))