Average Error: 7.9 → 4.7
Time: 6.9s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{\sqrt{x0}}{\frac{1 - x1}{\sqrt{x0}}} - x0 \cdot x0}\right)}{\sqrt[3]{{\left(\frac{x0}{1 - x1}\right)}^{3}} + x0}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{\sqrt{x0}}{\frac{1 - x1}{\sqrt{x0}}} - x0 \cdot x0}\right)}{\sqrt[3]{{\left(\frac{x0}{1 - x1}\right)}^{3}} + x0}
double f(double x0, double x1) {
        double r159480 = x0;
        double r159481 = 1.0;
        double r159482 = x1;
        double r159483 = r159481 - r159482;
        double r159484 = r159480 / r159483;
        double r159485 = r159484 - r159480;
        return r159485;
}
↓
double f(double x0, double x1) {
        double r159486 = x0;
        double r159487 = 1.0;
        double r159488 = x1;
        double r159489 = r159487 - r159488;
        double r159490 = r159486 / r159489;
        double r159491 = sqrt(r159486);
        double r159492 = r159489 / r159491;
        double r159493 = r159491 / r159492;
        double r159494 = r159490 * r159493;
        double r159495 = r159486 * r159486;
        double r159496 = r159494 - r159495;
        double r159497 = exp(r159496);
        double r159498 = log(r159497);
        double r159499 = 3.0;
        double r159500 = pow(r159490, r159499);
        double r159501 = cbrt(r159500);
        double r159502 = r159501 + r159486;
        double r159503 = r159498 / r159502;
        return r159503;
}

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
Target0.3
Herbie4.7
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

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

  3. Applied flip--7.3

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

  5. Applied add-sqr-sqrt7.3

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

  6. Applied associate-/l*5.6

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

  8. Applied add-log-exp5.6

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

  9. Applied add-log-exp5.6

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

  10. Applied diff-log5.5

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

  11. Simplified4.7

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

  13. Applied add-cbrt-cube4.7

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

  14. Applied add-cbrt-cube4.7

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

  15. Applied cbrt-undiv4.7

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

  16. Simplified4.7

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

  17. Final simplification4.7

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

Reproduce

herbie shell --seed 2020047 
(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))