Average Error: 8.4 → 5.5
Time: 3.9s
Precision: binary64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\left(\left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} + {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right) \cdot \left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} - {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right)\right) \cdot \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}}{\left(\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\left(\left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} + {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right) \cdot \left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} - {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right)\right) \cdot \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}}{\left(\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}
double code(double x0, double x1) {
	return ((double) (((double) (x0 / ((double) (1.0 - x1)))) - x0));
}
↓
double code(double x0, double x1) {
	return ((double) (((double) (((double) (((double) (((double) pow(((double) sqrt(((double) (x0 / ((double) (1.0 - x1)))))), 1.5)) + ((double) pow(((double) sqrt(x0)), 1.5)))) * ((double) (((double) pow(((double) sqrt(((double) (x0 / ((double) (1.0 - x1)))))), 1.5)) - ((double) pow(((double) sqrt(x0)), 1.5)))))) * ((double) (((double) (((double) (((double) (x0 / ((double) (1.0 - x1)))) * ((double) (x0 / ((double) (1.0 - x1)))))) - ((double) (x0 * x0)))) / ((double) (x0 + ((double) (x0 / ((double) (1.0 - x1)))))))))) / ((double) (((double) (((double) (((double) (x0 / ((double) (1.0 - x1)))) + x0)) + ((double) (((double) sqrt(((double) (x0 / ((double) (1.0 - x1)))))) * ((double) sqrt(x0)))))) * ((double) (((double) sqrt(((double) (x0 / ((double) (1.0 - x1)))))) - ((double) sqrt(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
Herbie5.5
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 8.4

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

  3. Applied add-sqr-sqrt8.4

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

  4. Applied add-sqr-sqrt7.6

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

  5. Applied difference-of-squares7.7

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

  7. Applied flip3--7.9

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

  8. Applied flip-+8.0

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

  9. Applied frac-times8.0

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

  10. Simplified7.5

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

  11. Simplified7.5

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

  13. Applied flip-+6.0

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

  14. Simplified6.0

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

  15. Simplified6.0

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

  17. Applied sqr-pow6.0

    \[\leadsto \frac{\left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{3} - \color{blue}{{\left(\sqrt{x0}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt{x0}\right)}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}}{\left(\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]

  18. Applied sqr-pow4.0

    \[\leadsto \frac{\left(\color{blue}{{\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\left(\frac{3}{2}\right)}} - {\left(\sqrt{x0}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt{x0}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}}{\left(\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]

  19. Applied difference-of-squares5.5

    \[\leadsto \frac{\color{blue}{\left(\left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\left(\frac{3}{2}\right)} + {\left(\sqrt{x0}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\left(\frac{3}{2}\right)} - {\left(\sqrt{x0}\right)}^{\left(\frac{3}{2}\right)}\right)\right)} \cdot \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}}{\left(\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]

  20. Simplified5.5

    \[\leadsto \frac{\left(\color{blue}{\left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} + {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right)} \cdot \left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\left(\frac{3}{2}\right)} - {\left(\sqrt{x0}\right)}^{\left(\frac{3}{2}\right)}\right)\right) \cdot \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}}{\left(\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]

  21. Simplified5.5

    \[\leadsto \frac{\left(\left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} + {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right) \cdot \color{blue}{\left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} - {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right)}\right) \cdot \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}}{\left(\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]

  22. Final simplification5.5

    \[\leadsto \frac{\left(\left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} + {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right) \cdot \left({\left(\sqrt{\frac{x0}{1 - x1}}\right)}^{\frac{3}{2}} - {\left(\sqrt{x0}\right)}^{\frac{3}{2}}\right)\right) \cdot \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}}{\left(\left(\frac{x0}{1 - x1} + x0\right) + \sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{x0}\right) \cdot \left(\sqrt{\frac{x0}{1 - x1}} - \sqrt{x0}\right)}\]

Reproduce

herbie shell --seed 2020161 
(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) (- 1.0 x1))

  (- (/ x0 (- 1.0 x1)) x0))