Average Error: 7.8 → 4.7
Time: 3.8s
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 r171031 = x0;
        double r171032 = 1.0;
        double r171033 = x1;
        double r171034 = r171032 - r171033;
        double r171035 = r171031 / r171034;
        double r171036 = r171035 - r171031;
        return r171036;
}
↓
double f(double x0, double x1) {
        double r171037 = x0;
        double r171038 = 1.0;
        double r171039 = 3.0;
        double r171040 = pow(r171038, r171039);
        double r171041 = x1;
        double r171042 = pow(r171041, r171039);
        double r171043 = r171040 - r171042;
        double r171044 = r171037 / r171043;
        double r171045 = r171038 * r171038;
        double r171046 = r171041 * r171041;
        double r171047 = r171038 * r171041;
        double r171048 = r171046 + r171047;
        double r171049 = r171045 + r171048;
        double r171050 = r171044 * r171049;
        double r171051 = r171045 - r171046;
        double r171052 = r171037 / r171051;
        double r171053 = r171038 + r171041;
        double r171054 = r171052 * r171053;
        double r171055 = r171050 * r171054;
        double r171056 = r171037 * r171037;
        double r171057 = r171055 - r171056;
        double r171058 = exp(r171057);
        double r171059 = log(r171058);
        double r171060 = r171038 - r171041;
        double r171061 = r171037 / r171060;
        double r171062 = r171061 + r171037;
        double r171063 = r171059 / r171062;
        return r171063;
}

Error

Bits error versus x0

Bits error versus x1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.8
Target0.2
Herbie4.7
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.8

    \[\frac{x0}{1 - x1} - x0\]
  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 2020024 
(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))