(- (/ x0 (- 1 x1)) x0)

Average Error: 8.4 → 4.3

Time: 3.4s

Precision: 64

\[x0 = 1.855 \land x1 = 2.09000000000000012 \cdot 10^{-4} \lor x0 = 2.98499999999999988 \land x1 = 0.018599999999999998\]

Math

\[\frac{x0}{1 - x1} - x0\]

↓

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

C

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) log(((double) exp(((double) (((double) (((double) (1.0 / ((double) (1.0 - x1)))) / ((double) (((double) (1.0 - x1)) / x0)))) - x0)))))))) / ((double) (((double) (x0 / ((double) (1.0 - x1)))) + x0))));
	}
	return VAR;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	8.4
Target	0.5
Herbie	4.3

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

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-inv 9.1

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

   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-inv 4.0

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

   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 clear-num 2.8

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

   11. Applied add-log-exp 2.8

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

   12. Applied add-log-exp 2.8

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

   13. Applied diff-log 3.9

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

   14. Simplified 1.0

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

4. Final simplification 4.3

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

Reproduce

herbie shell --seed 2020114 
(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))