Average Error: 7.8 → 5.6
Time: 3.4s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\frac{\frac{x0}{\sqrt{1 - x1}}}{\sqrt{1 - x1}} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\left(\sqrt[3]{\frac{x0}{1 - x1} + x0} \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}\right) \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\frac{\frac{x0}{\sqrt{1 - x1}}}{\sqrt{1 - x1}} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\left(\sqrt[3]{\frac{x0}{1 - x1} + x0} \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}\right) \cdot \sqrt[3]{\frac{x0}{1 - x1} + x0}}
double f(double x0, double x1) {
        double r189824 = x0;
        double r189825 = 1.0;
        double r189826 = x1;
        double r189827 = r189825 - r189826;
        double r189828 = r189824 / r189827;
        double r189829 = r189828 - r189824;
        return r189829;
}
↓
double f(double x0, double x1) {
        double r189830 = x0;
        double r189831 = 1.0;
        double r189832 = x1;
        double r189833 = r189831 - r189832;
        double r189834 = sqrt(r189833);
        double r189835 = r189830 / r189834;
        double r189836 = r189835 / r189834;
        double r189837 = r189830 / r189833;
        double r189838 = r189836 * r189837;
        double r189839 = r189830 * r189830;
        double r189840 = r189838 - r189839;
        double r189841 = r189837 + r189830;
        double r189842 = cbrt(r189841);
        double r189843 = r189842 * r189842;
        double r189844 = r189843 * r189842;
        double r189845 = r189840 / r189844;
        return r189845;
}

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
Herbie5.6
\[\frac{x0 \cdot x1}{1 - x1}\]

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 add-sqr-sqrt5.6

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

  6. Applied associate-/r*5.6

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

  8. Applied add-cube-cbrt5.6

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

  9. Final simplification5.6

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

Reproduce

herbie shell --seed 2020056 
(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))