Average Error: 7.8 → 6.3
Time: 3.6s
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 r183444 = x0;
        double r183445 = 1.0;
        double r183446 = x1;
        double r183447 = r183445 - r183446;
        double r183448 = r183444 / r183447;
        double r183449 = r183448 - r183444;
        return r183449;
}
↓
double f(double x0, double x1) {
        double r183450 = 1.0;
        double r183451 = x0;
        double r183452 = exp(r183451);
        double r183453 = sqrt(r183452);
        double r183454 = r183450 / r183453;
        double r183455 = log(r183454);
        double r183456 = cbrt(r183451);
        double r183457 = 1.0;
        double r183458 = x1;
        double r183459 = r183457 - r183458;
        double r183460 = r183456 / r183459;
        double r183461 = 0.6666666666666666;
        double r183462 = pow(r183451, r183461);
        double r183463 = fma(r183460, r183462, r183455);
        double r183464 = r183455 + r183463;
        double r183465 = log(r183464);
        double r183466 = exp(r183465);
        return r183466;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original7.8
Target0.3
Herbie6.3
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.8

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

  3. Applied *-un-lft-identity7.8

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

  4. Applied add-cube-cbrt7.8

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

  5. Applied times-frac8.1

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

  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-exp7.7

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

  11. Simplified6.6

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

  13. Applied add-sqr-sqrt7.1

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

  14. Applied *-un-lft-identity7.1

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

  15. Applied unpow-prod-down7.1

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

  16. Applied times-frac6.7

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

  17. Applied log-prod7.0

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

  18. Simplified7.0

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

  19. Simplified6.3

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

  20. Final simplification6.3

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

Reproduce

herbie shell --seed 2020089 +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))