Average Error: 7.9 → 4.7
Time: 3.5s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\log \left(e^{\left(\frac{x0}{{1}^{3} - {x1}^{3}} \cdot \left(1 \cdot 1 + \left(x1 \cdot x1 + 1 \cdot x1\right)\right)\right) \cdot \left(\frac{x0}{1 \cdot 1 - x1 \cdot x1} \cdot \left(1 + x1\right)\right) - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\log \left(e^{\left(\frac{x0}{{1}^{3} - {x1}^{3}} \cdot \left(1 \cdot 1 + \left(x1 \cdot x1 + 1 \cdot x1\right)\right)\right) \cdot \left(\frac{x0}{1 \cdot 1 - x1 \cdot x1} \cdot \left(1 + x1\right)\right) - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}
double f(double x0, double x1) {
        double r205660 = x0;
        double r205661 = 1.0;
        double r205662 = x1;
        double r205663 = r205661 - r205662;
        double r205664 = r205660 / r205663;
        double r205665 = r205664 - r205660;
        return r205665;
}
↓
double f(double x0, double x1) {
        double r205666 = x0;
        double r205667 = 1.0;
        double r205668 = 3.0;
        double r205669 = pow(r205667, r205668);
        double r205670 = x1;
        double r205671 = pow(r205670, r205668);
        double r205672 = r205669 - r205671;
        double r205673 = r205666 / r205672;
        double r205674 = r205667 * r205667;
        double r205675 = r205670 * r205670;
        double r205676 = r205667 * r205670;
        double r205677 = r205675 + r205676;
        double r205678 = r205674 + r205677;
        double r205679 = r205673 * r205678;
        double r205680 = r205674 - r205675;
        double r205681 = r205666 / r205680;
        double r205682 = r205667 + r205670;
        double r205683 = r205681 * r205682;
        double r205684 = r205679 * r205683;
        double r205685 = r205666 * r205666;
        double r205686 = r205684 - r205685;
        double r205687 = exp(r205686);
        double r205688 = log(r205687);
        double r205689 = r205667 - r205670;
        double r205690 = r205666 / r205689;
        double r205691 = r205690 + r205666;
        double r205692 = r205688 / r205691;
        return r205692;
}

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. 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 flip--5.6

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

  6. Applied associate-/r/6.2

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

  8. Applied add-log-exp6.2

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

  9. Applied add-log-exp6.2

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

  10. Applied diff-log5.8

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

  11. Simplified5.8

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

  13. Applied flip3--5.8

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

  14. Applied associate-/r/4.7

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

  15. Final simplification4.7

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

Reproduce

herbie shell --seed 2020033 
(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))