Average Error: 8.4 → 4.2
Time: 2.7s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{x0 \cdot \frac{\frac{\mathsf{fma}\left({\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3} \cdot {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3}, {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3}, -{\left({x0}^{3}\right)}^{3}\right)}{\mathsf{fma}\left({x0}^{3}, {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3} + {x0}^{3}, {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{6}\right)}}{\mathsf{fma}\left(\frac{\frac{x0}{1 - x1}}{1 - x1}, \frac{\frac{x0}{1 - x1}}{1 - x1} + x0, x0 \cdot x0\right)}}{\frac{x0}{1 - x1} + x0}\]
\frac{x0}{1 - x1} - x0
↓
\frac{x0 \cdot \frac{\frac{\mathsf{fma}\left({\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3} \cdot {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3}, {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3}, -{\left({x0}^{3}\right)}^{3}\right)}{\mathsf{fma}\left({x0}^{3}, {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3} + {x0}^{3}, {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{6}\right)}}{\mathsf{fma}\left(\frac{\frac{x0}{1 - x1}}{1 - x1}, \frac{\frac{x0}{1 - x1}}{1 - x1} + x0, x0 \cdot x0\right)}}{\frac{x0}{1 - x1} + x0}
double code(double x0, double x1) {
	return ((x0 / (1.0 - x1)) - x0);
}
↓
double code(double x0, double x1) {
	return ((x0 * ((fma((pow(((x0 / (1.0 - x1)) / (1.0 - x1)), 3.0) * pow(((x0 / (1.0 - x1)) / (1.0 - x1)), 3.0)), pow(((x0 / (1.0 - x1)) / (1.0 - x1)), 3.0), -pow(pow(x0, 3.0), 3.0)) / fma(pow(x0, 3.0), (pow(((x0 / (1.0 - x1)) / (1.0 - x1)), 3.0) + pow(x0, 3.0)), pow(((x0 / (1.0 - x1)) / (1.0 - x1)), 6.0))) / fma(((x0 / (1.0 - x1)) / (1.0 - x1)), (((x0 / (1.0 - x1)) / (1.0 - x1)) + x0), (x0 * x0)))) / ((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
Herbie4.2
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 8.4

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

  3. Applied flip--7.7

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

  4. Simplified6.9

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

  6. Applied flip3--6.1

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

  7. Simplified6.1

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

  9. Applied flip3--5.3

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

  10. Simplified5.3

    \[\leadsto \frac{x0 \cdot \frac{\frac{{\left({\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3}\right)}^{3} - {\left({x0}^{3}\right)}^{3}}{\color{blue}{\mathsf{fma}\left({x0}^{3}, {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{3} + {x0}^{3}, {\left(\frac{\frac{x0}{1 - x1}}{1 - x1}\right)}^{6}\right)}}}{\mathsf{fma}\left(\frac{\frac{x0}{1 - x1}}{1 - x1}, \frac{\frac{x0}{1 - x1}}{1 - x1} + x0, x0 \cdot x0\right)}}{\frac{x0}{1 - x1} + x0}\]
  11. Using strategy rm

  12. Applied unpow35.3

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

  13. Applied fma-neg4.2

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

  14. Final simplification4.2

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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))