Average Error: 7.9 → 4.7
Time: 3.5s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \left(\frac{\sqrt{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt{x0}}{\sqrt{1 - x1}}\right) - x0 \cdot x0}\right)}{\frac{\frac{x0}{\sqrt{1} + \sqrt{x1}}}{\sqrt{1} - \sqrt{x1}} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\
\;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \left(\frac{\sqrt{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt{x0}}{\sqrt{1 - x1}}\right) - x0 \cdot x0}\right)}{\frac{\frac{x0}{\sqrt{1} + \sqrt{x1}}}{\sqrt{1} - \sqrt{x1}} + x0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\

\end{array}
double f(double x0, double x1) {
        double r165608 = x0;
        double r165609 = 1.0;
        double r165610 = x1;
        double r165611 = r165609 - r165610;
        double r165612 = r165608 / r165611;
        double r165613 = r165612 - r165608;
        return r165613;
}
↓
double f(double x0, double x1) {
        double r165614 = x1;
        double r165615 = 0.00021208908081054686;
        bool r165616 = r165614 <= r165615;
        double r165617 = x0;
        double r165618 = 1.0;
        double r165619 = r165618 - r165614;
        double r165620 = r165617 / r165619;
        double r165621 = sqrt(r165617);
        double r165622 = sqrt(r165619);
        double r165623 = r165621 / r165622;
        double r165624 = r165623 * r165623;
        double r165625 = r165620 * r165624;
        double r165626 = r165617 * r165617;
        double r165627 = r165625 - r165626;
        double r165628 = exp(r165627);
        double r165629 = log(r165628);
        double r165630 = sqrt(r165618);
        double r165631 = sqrt(r165614);
        double r165632 = r165630 + r165631;
        double r165633 = r165617 / r165632;
        double r165634 = r165630 - r165631;
        double r165635 = r165633 / r165634;
        double r165636 = r165635 + r165617;
        double r165637 = r165629 / r165636;
        double r165638 = r165620 * r165620;
        double r165639 = r165638 - r165626;
        double r165640 = exp(r165639);
        double r165641 = log(r165640);
        double r165642 = r165620 + r165617;
        double r165643 = r165641 / r165642;
        double r165644 = r165616 ? r165637 : r165643;
        return r165644;
}

Error

Bits error versus x0

Bits error versus x1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.9
Target0.2
Herbie4.7
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Split input into 2 regimes

  2. if x1 < 0.00021208908081054686

    1. Initial program 11.2

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

    3. Applied flip--11.4

      \[\leadsto \color{blue}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt8.1

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

    6. Applied add-sqr-sqrt8.1

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

    7. Applied times-frac8.1

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

    9. Applied add-log-exp8.1

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

    10. Applied add-log-exp8.1

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

    11. Applied diff-log7.4

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

    12. Simplified7.4

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

    14. Applied add-sqr-sqrt7.4

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

    15. Applied add-sqr-sqrt7.4

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

    16. Applied difference-of-squares7.4

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

    17. Applied associate-/r*7.4

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

    if 0.00021208908081054686 < x1

    1. Initial program 4.5

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

    3. Applied flip--3.2

      \[\leadsto \color{blue}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}}\]
    4. Using strategy rm

    5. Applied add-log-exp3.2

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

    6. Applied add-log-exp3.2

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

    7. Applied diff-log3.5

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

    8. Simplified1.9

      \[\leadsto \frac{\log \color{blue}{\left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}}{\frac{x0}{1 - x1} + x0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \left(\frac{\sqrt{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt{x0}}{\sqrt{1 - x1}}\right) - x0 \cdot x0}\right)}{\frac{\frac{x0}{\sqrt{1} + \sqrt{x1}}}{\sqrt{1} - \sqrt{x1}} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))