Average Error: 8.4 → 5.1
Time: 3.1s
Precision: 64
\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;x1 \le 0.0182045976562499982:\\ \;\;\;\;\frac{x0 \cdot \left(\frac{1}{1 - x1} \cdot \frac{x0}{1 - x1} - x0\right)}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} - x0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \left(\frac{1}{1 - x1} \cdot \left(\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}}\right) - x0\right)}{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}} + x0}\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;x1 \le 0.0182045976562499982:\\
\;\;\;\;\frac{x0 \cdot \left(\frac{1}{1 - x1} \cdot \frac{x0}{1 - x1} - x0\right)}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} - x0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x0 \cdot \left(\frac{1}{1 - x1} \cdot \left(\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}}\right) - x0\right)}{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}} + x0}\\

\end{array}
double code(double x0, double x1) {
	return ((double) (((double) (x0 / ((double) (1.0 - x1)))) - x0));
}
↓
double code(double x0, double x1) {
	double VAR;
	if ((x1 <= 0.018204597656249998)) {
		VAR = ((double) (((double) (x0 * ((double) (((double) (((double) (1.0 / ((double) (1.0 - x1)))) * ((double) (x0 / ((double) (1.0 - x1)))))) - x0)))) / ((double) (((double) (((double) (((double) (x0 / ((double) (1.0 - x1)))) * ((double) (x0 / ((double) (1.0 - x1)))))) - ((double) (x0 * x0)))) / ((double) (((double) (x0 / ((double) (1.0 - x1)))) - x0))))));
	} else {
		VAR = ((double) (((double) (x0 * ((double) (((double) (((double) (1.0 / ((double) (1.0 - x1)))) * ((double) (((double) sqrt(((double) (x0 / ((double) (1.0 - x1)))))) * ((double) sqrt(((double) (x0 / ((double) (1.0 - x1)))))))))) - x0)))) / ((double) (((double) (((double) sqrt(((double) (x0 / ((double) (1.0 - x1)))))) * ((double) sqrt(((double) (x0 / ((double) (1.0 - x1)))))))) + x0))));
	}
	return VAR;
}

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

Derivation

  1. Split input into 2 regimes

  2. if x1 < 0.018204597656249998

    1. Initial program 11.3

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

    3. Applied flip--11.4

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

    5. Applied div-inv9.1

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

    6. Applied associate-*l*9.1

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

    7. Applied distribute-lft-out--9.1

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

    9. Applied flip-+7.7

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

    if 0.018204597656249998 < x1

    1. Initial program 5.5

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

    3. Applied flip--4.0

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

    5. Applied div-inv4.0

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

    6. Applied associate-*l*4.0

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

    7. Applied distribute-lft-out--4.7

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

    9. Applied add-sqr-sqrt2.8

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

    11. Applied add-sqr-sqrt2.6

      \[\leadsto \frac{x0 \cdot \left(\frac{1}{1 - x1} \cdot \left(\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}}\right) - x0\right)}{\color{blue}{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}}} + x0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 0.0182045976562499982:\\ \;\;\;\;\frac{x0 \cdot \left(\frac{1}{1 - x1} \cdot \frac{x0}{1 - x1} - x0\right)}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} - x0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \left(\frac{1}{1 - x1} \cdot \left(\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}}\right) - x0\right)}{\sqrt{\frac{x0}{1 - x1}} \cdot \sqrt{\frac{x0}{1 - x1}} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(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))