Average Error: 8.4 → 5.7
Time: 3.6s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\frac{\frac{{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3}\right)}^{3} - {\left({\left(x0 \cdot x0\right)}^{3}\right)}^{3}}{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{6} + {\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3} \cdot {\left(x0 \cdot x0\right)}^{3}\right) + {x0}^{12}}}{\frac{x0}{1 - x1} \cdot \left({\left(\frac{x0}{1 - x1}\right)}^{3} + \frac{{x0}^{3}}{1 - x1}\right) + {x0}^{4}}}{\frac{x0}{1 - x1} + x0}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\frac{\frac{{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3}\right)}^{3} - {\left({\left(x0 \cdot x0\right)}^{3}\right)}^{3}}{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{6} + {\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3} \cdot {\left(x0 \cdot x0\right)}^{3}\right) + {x0}^{12}}}{\frac{x0}{1 - x1} \cdot \left({\left(\frac{x0}{1 - x1}\right)}^{3} + \frac{{x0}^{3}}{1 - x1}\right) + {x0}^{4}}}{\frac{x0}{1 - x1} + x0}
double code(double x0, double x1) {
	return ((x0 / (1.0 - x1)) - x0);
}
↓
double code(double x0, double x1) {
	return ((((pow(pow(((x0 / (1.0 - x1)) * (x0 * (1.0 / (1.0 - x1)))), 3.0), 3.0) - pow(pow((x0 * x0), 3.0), 3.0)) / ((pow(((x0 / (1.0 - x1)) * (x0 * (1.0 / (1.0 - x1)))), 6.0) + (pow(((x0 / (1.0 - x1)) * (x0 * (1.0 / (1.0 - x1)))), 3.0) * pow((x0 * x0), 3.0))) + pow(x0, 12.0))) / (((x0 / (1.0 - x1)) * (pow((x0 / (1.0 - x1)), 3.0) + (pow(x0, 3.0) / (1.0 - x1)))) + pow(x0, 4.0))) / ((x0 / (1.0 - x1)) + x0));
}

Error

Bits error versus x0

Bits error versus x1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.4
Target0.5
Herbie5.7
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 8.4

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

  3. Applied flip--7.8

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

  5. Applied div-inv6.6

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

  7. Applied flip3--6.0

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

  8. Simplified6.0

    \[\leadsto \frac{\frac{{\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3} - {\left(x0 \cdot x0\right)}^{3}}{\color{blue}{\frac{x0}{1 - x1} \cdot \left({\left(\frac{x0}{1 - x1}\right)}^{3} + \frac{{x0}^{3}}{1 - x1}\right) + {x0}^{4}}}}{\frac{x0}{1 - x1} + x0}\]
  9. Using strategy rm

  10. Applied flip3--5.6

    \[\leadsto \frac{\frac{\color{blue}{\frac{{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3}\right)}^{3} - {\left({\left(x0 \cdot x0\right)}^{3}\right)}^{3}}{{\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3} \cdot {\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3} + \left({\left(x0 \cdot x0\right)}^{3} \cdot {\left(x0 \cdot x0\right)}^{3} + {\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3} \cdot {\left(x0 \cdot x0\right)}^{3}\right)}}}{\frac{x0}{1 - x1} \cdot \left({\left(\frac{x0}{1 - x1}\right)}^{3} + \frac{{x0}^{3}}{1 - x1}\right) + {x0}^{4}}}{\frac{x0}{1 - x1} + x0}\]

  11. Simplified5.7

    \[\leadsto \frac{\frac{\frac{{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3}\right)}^{3} - {\left({\left(x0 \cdot x0\right)}^{3}\right)}^{3}}{\color{blue}{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{6} + {\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3} \cdot {\left(x0 \cdot x0\right)}^{3}\right) + {x0}^{12}}}}{\frac{x0}{1 - x1} \cdot \left({\left(\frac{x0}{1 - x1}\right)}^{3} + \frac{{x0}^{3}}{1 - x1}\right) + {x0}^{4}}}{\frac{x0}{1 - x1} + x0}\]

  12. Final simplification5.7

    \[\leadsto \frac{\frac{\frac{{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3}\right)}^{3} - {\left({\left(x0 \cdot x0\right)}^{3}\right)}^{3}}{\left({\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{6} + {\left(\frac{x0}{1 - x1} \cdot \left(x0 \cdot \frac{1}{1 - x1}\right)\right)}^{3} \cdot {\left(x0 \cdot x0\right)}^{3}\right) + {x0}^{12}}}{\frac{x0}{1 - x1} \cdot \left({\left(\frac{x0}{1 - x1}\right)}^{3} + \frac{{x0}^{3}}{1 - x1}\right) + {x0}^{4}}}{\frac{x0}{1 - x1} + x0}\]

Reproduce

herbie shell --seed 2020102 
(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))