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

Average Error: 7.8 → 4.6

Time: 3.5s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 - x1 \le 0.99059549999999996:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\left(\frac{\sqrt{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt{x0}}{\sqrt{1 - x1}}\right) \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\left(\sqrt[3]{\frac{x0}{1 - x1}} \cdot \sqrt[3]{\frac{x0}{1 - x1}}\right) \cdot \sqrt[3]{\frac{x0}{1 - x1}} + x0}\\ \end{array}\]

C

double code(double x0, double x1) {
	return ((x0 / (1.0 - x1)) - x0);
}

↓

double code(double x0, double x1) {
	double temp;
	if (((1.0 - x1) <= 0.9905955)) {
		temp = (log(exp((((x0 / (1.0 - x1)) * (x0 / (1.0 - x1))) - (x0 * x0)))) / ((x0 / (1.0 - x1)) + x0));
	} else {
		temp = (log(exp(((((sqrt(x0) / sqrt((1.0 - x1))) * (sqrt(x0) / sqrt((1.0 - x1)))) * (x0 / (1.0 - x1))) - (x0 * x0)))) / (((cbrt((x0 / (1.0 - x1))) * cbrt((x0 / (1.0 - x1)))) * cbrt((x0 / (1.0 - x1)))) + x0));
	}
	return temp;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.2
Herbie	4.6

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

Derivation

1. Split input into 2 regimes

2. if (- 1.0 x1) < 0.9905955

   1. Initial program 4.5

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

   3. Applied flip-- 3.2

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

   5. Applied add-log-exp 3.2

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

   6. Applied add-log-exp 3.2

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

   7. Applied diff-log 3.5

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

   8. Simplified 1.9

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

   if 0.9905955 < (- 1.0 x1)

   1. Initial program 11.2

      \[\frac{x0}{1 - x1} - x0\]
   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 add-sqr-sqrt 8.1

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

   6. Applied add-sqr-sqrt 8.1

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

   7. Applied times-frac 8.1

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

   9. Applied add-log-exp 8.1

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

   10. Applied add-log-exp 8.1

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

   11. Applied diff-log 7.4

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

   12. Simplified 7.4

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

   14. Applied add-cube-cbrt 7.4

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

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x1 \le 0.99059549999999996:\\ \;\;\;\;\frac{\log \left(e^{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\left(\frac{\sqrt{x0}}{\sqrt{1 - x1}} \cdot \frac{\sqrt{x0}}{\sqrt{1 - x1}}\right) \cdot \frac{x0}{1 - x1} - x0 \cdot x0}\right)}{\left(\sqrt[3]{\frac{x0}{1 - x1}} \cdot \sqrt[3]{\frac{x0}{1 - x1}}\right) \cdot \sqrt[3]{\frac{x0}{1 - x1}} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))