Average Error: 7.9 → 4.7
Time: 3.1s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt[3]{x0}}{\sqrt{1 - x1}}\right) - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]
\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[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt[3]{x0}}{\sqrt{1 - x1}}\right) - x0 \cdot x0}\right)}{\frac{x0}{1 - 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 r194571 = x0;
        double r194572 = 1.0;
        double r194573 = x1;
        double r194574 = r194572 - r194573;
        double r194575 = r194571 / r194574;
        double r194576 = r194575 - r194571;
        return r194576;
}
↓
double f(double x0, double x1) {
        double r194577 = x1;
        double r194578 = 0.00021208908081054686;
        bool r194579 = r194577 <= r194578;
        double r194580 = x0;
        double r194581 = 1.0;
        double r194582 = r194581 - r194577;
        double r194583 = r194580 / r194582;
        double r194584 = cbrt(r194580);
        double r194585 = r194584 * r194584;
        double r194586 = sqrt(r194582);
        double r194587 = r194585 / r194586;
        double r194588 = r194584 / r194586;
        double r194589 = r194587 * r194588;
        double r194590 = r194583 * r194589;
        double r194591 = r194580 * r194580;
        double r194592 = r194590 - r194591;
        double r194593 = exp(r194592);
        double r194594 = log(r194593);
        double r194595 = r194583 + r194580;
        double r194596 = r194594 / r194595;
        double r194597 = r194583 * r194583;
        double r194598 = r194597 - r194591;
        double r194599 = exp(r194598);
        double r194600 = log(r194599);
        double r194601 = r194600 / r194595;
        double r194602 = r194579 ? r194596 : r194601;
        return r194602;
}

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-sqrt8.0

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

    6. Applied add-cube-cbrt8.0

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

    7. Applied times-frac8.0

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

    9. Applied add-log-exp8.0

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

    10. Applied add-log-exp8.0

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

    11. Applied diff-log7.3

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

    12. Simplified7.3

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

    if 0.00021208908081054686 < x1

    1. Initial program 4.5

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

    3. Applied flip--3.1

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

    5. Applied add-log-exp3.1

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

    6. Applied add-log-exp3.1

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt[3]{x0}}{\sqrt{1 - x1}}\right) - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(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))