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

Average Error: 8.3 → 3.3

Time: 2.2s

Precision: binary64

\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \leq 0.00021208908081054686:\\ \;\;\;\;\frac{x0 \cdot \left(\log \left(\sqrt[3]{e^{\frac{x0}{1 + x1 \cdot \left(x1 - 2\right)} - x0}}\right) + \log \left(\sqrt[3]{e^{\frac{x0}{1 + x1 \cdot \left(x1 - 2\right)} - x0}}\right) \cdot 2\right)}{x0 + \frac{x0}{1 - x1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \log \left(e^{\frac{x0}{1 + x1 \cdot \left(x1 - 2\right)} - x0}\right)}{x0 + \frac{1}{\frac{1 - x1}{x0}}}\\ \end{array}\]

FPCore

(FPCore (x0 x1) :precision binary64 (- (/ x0 (- 1.0 x1)) x0))

↓

(FPCore (x0 x1)
 :precision binary64
 (if (<= x1 0.00021208908081054686)
   (/
    (*
     x0
     (+
      (log (cbrt (exp (- (/ x0 (+ 1.0 (* x1 (- x1 2.0)))) x0))))
      (* (log (cbrt (exp (- (/ x0 (+ 1.0 (* x1 (- x1 2.0)))) x0)))) 2.0)))
    (+ x0 (/ x0 (- 1.0 x1))))
   (/
    (* x0 (log (exp (- (/ x0 (+ 1.0 (* x1 (- x1 2.0)))) x0))))
    (+ x0 (/ 1.0 (/ (- 1.0 x1) x0))))))

C

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

↓

double code(double x0, double x1) {
	double tmp;
	if ((x1 <= 0.00021208908081054686)) {
		tmp = (((double) (x0 * ((double) (((double) log(((double) cbrt(((double) exp(((double) ((x0 / ((double) (1.0 + ((double) (x1 * ((double) (x1 - 2.0))))))) - x0)))))))) + ((double) (((double) log(((double) cbrt(((double) exp(((double) ((x0 / ((double) (1.0 + ((double) (x1 * ((double) (x1 - 2.0))))))) - x0)))))))) * 2.0)))))) / ((double) (x0 + (x0 / ((double) (1.0 - x1))))));
	} else {
		tmp = (((double) (x0 * ((double) log(((double) exp(((double) ((x0 / ((double) (1.0 + ((double) (x1 * ((double) (x1 - 2.0))))))) - x0)))))))) / ((double) (x0 + (1.0 / (((double) (1.0 - x1)) / x0)))));
	}
	return tmp;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	8.3
Target	0.5
Herbie	3.3

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

Derivation

1. Split input into 2 regimes

2. if x1 < 2.12089080810546861e-4

   1. Initial program Error: 11.3 bits

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

   3. Applied flip-- Error: 11.4 bits

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

   4. Simplified Error: 9.1 bits

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

   5. Simplified Error: 9.1 bits

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

   6. Taylor expanded around 0 Error: 11.3 bits

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

   7. Simplified Error: 9.1 bits

      \[\leadsto \frac{x0 \cdot \left(\frac{x0}{\color{blue}{1 + x1 \cdot \left(x1 - 2\right)}} - x0\right)}{x0 + \frac{x0}{1 - x1}}\]
   8. Using strategy rm

   9. Applied add-log-exp Error: 9.1 bits

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

   10. Applied add-log-exp Error: 9.1 bits

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

   11. Applied diff-log Error: 8.9 bits

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

   12. Simplified Error: 8.9 bits

       \[\leadsto \frac{x0 \cdot \log \color{blue}{\left(e^{\frac{x0}{1 + x1 \cdot \left(x1 - 2\right)} - x0}\right)}}{x0 + \frac{x0}{1 - x1}}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt Error: 10.6 bits

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

   15. Applied log-prod Error: 9.9 bits

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

   16. Simplified Error: 6.6 bits

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

   if 2.12089080810546861e-4 < x1

   1. Initial program Error: 5.5 bits

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

   3. Applied flip-- Error: 4.0 bits

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

   4. Simplified Error: 4.7 bits

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

   5. Simplified Error: 4.7 bits

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

   6. Taylor expanded around 0 Error: 2.8 bits

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

   7. Simplified Error: 2.8 bits

      \[\leadsto \frac{x0 \cdot \left(\frac{x0}{\color{blue}{1 + x1 \cdot \left(x1 - 2\right)}} - x0\right)}{x0 + \frac{x0}{1 - x1}}\]
   8. Using strategy rm

   9. Applied add-log-exp Error: 2.8 bits

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

   10. Applied add-log-exp Error: 2.8 bits

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

   11. Applied diff-log Error: 3.9 bits

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

   12. Simplified Error: 1.0 bits

       \[\leadsto \frac{x0 \cdot \log \color{blue}{\left(e^{\frac{x0}{1 + x1 \cdot \left(x1 - 2\right)} - x0}\right)}}{x0 + \frac{x0}{1 - x1}}\]
   13. Using strategy rm

   14. Applied clear-num Error: 0 bits

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

4. Final simplification Error: 3.3 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \leq 0.00021208908081054686:\\ \;\;\;\;\frac{x0 \cdot \left(\log \left(\sqrt[3]{e^{\frac{x0}{1 + x1 \cdot \left(x1 - 2\right)} - x0}}\right) + \log \left(\sqrt[3]{e^{\frac{x0}{1 + x1 \cdot \left(x1 - 2\right)} - x0}}\right) \cdot 2\right)}{x0 + \frac{x0}{1 - x1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \log \left(e^{\frac{x0}{1 + x1 \cdot \left(x1 - 2\right)} - x0}\right)}{x0 + \frac{1}{\frac{1 - x1}{x0}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))