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

Average Error: 8.3 → 2.7

Time: 2.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;1 - x1 \leq 0.9814:\\ \;\;\;\;\sqrt[3]{x0} \cdot \frac{\left(\sqrt{{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}^{3} - {\left(\sqrt[3]{x0}\right)}^{3}} \cdot {x0}^{0.16666666666666666}\right) \cdot \left({x0}^{0.16666666666666666} \cdot \sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}}\right)}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} + \left(\sqrt[3]{x0} \cdot \sqrt[3]{x0} + \sqrt[3]{x0} \cdot \frac{\sqrt[3]{x0}}{1 - x1}\right)} \cdot \sqrt{\sqrt[3]{x0} + \frac{\sqrt[3]{x0}}{1 - x1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x0} \cdot \left(\sqrt[3]{x0} \cdot \left(\sqrt[3]{x0} \cdot \left(x1 + \left({x1}^{3} + \left(x1 \cdot x1 + {x1}^{4}\right)\right)\right)\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x0 x1)
 :precision binary64
 (if (<= (- 1.0 x1) 0.9814)
   (*
    (cbrt x0)
    (/
     (*
      (*
       (sqrt (- (pow (/ (cbrt x0) (- 1.0 x1)) 3.0) (pow (cbrt x0) 3.0)))
       (pow x0 0.16666666666666666))
      (*
       (pow x0 0.16666666666666666)
       (sqrt
        (-
         (* (/ (cbrt x0) (- 1.0 x1)) (/ (cbrt x0) (- 1.0 x1)))
         (* (cbrt x0) (cbrt x0))))))
     (*
      (sqrt
       (+
        (* (/ (cbrt x0) (- 1.0 x1)) (/ (cbrt x0) (- 1.0 x1)))
        (+ (* (cbrt x0) (cbrt x0)) (* (cbrt x0) (/ (cbrt x0) (- 1.0 x1))))))
      (sqrt (+ (cbrt x0) (/ (cbrt x0) (- 1.0 x1)))))))
   (*
    (cbrt x0)
    (*
     (cbrt x0)
     (* (cbrt x0) (+ x1 (+ (pow x1 3.0) (+ (* x1 x1) (pow x1 4.0)))))))))

C

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

↓

double code(double x0, double x1) {
	double tmp;
	if ((1.0 - x1) <= 0.9814) {
		tmp = cbrt(x0) * (((sqrt(pow((cbrt(x0) / (1.0 - x1)), 3.0) - pow(cbrt(x0), 3.0)) * pow(x0, 0.16666666666666666)) * (pow(x0, 0.16666666666666666) * sqrt(((cbrt(x0) / (1.0 - x1)) * (cbrt(x0) / (1.0 - x1))) - (cbrt(x0) * cbrt(x0))))) / (sqrt(((cbrt(x0) / (1.0 - x1)) * (cbrt(x0) / (1.0 - x1))) + ((cbrt(x0) * cbrt(x0)) + (cbrt(x0) * (cbrt(x0) / (1.0 - x1))))) * sqrt(cbrt(x0) + (cbrt(x0) / (1.0 - x1)))));
	} else {
		tmp = cbrt(x0) * (cbrt(x0) * (cbrt(x0) * (x1 + (pow(x1, 3.0) + ((x1 * x1) + pow(x1, 4.0))))));
	}
	return tmp;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	8.3
Target	0.5
Herbie	2.7

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 x1) < 0.98140000000000005

   1. Initial program 5.5

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

   3. Applied *-un-lft-identity_binary64 5.5

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

   4. Applied *-un-lft-identity_binary64 5.5

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

   5. Applied distribute-lft-out--_binary64 5.5

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

   6. Simplified 4.4

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

   8. Applied add-cube-cbrt_binary64 4.4

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

   9. Applied associate-*r*_binary64 4.5

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

   10. Simplified 2.6

       \[\leadsto 1 \cdot \left(\color{blue}{\left(\sqrt[3]{x0} \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}\right)\right)} \cdot \sqrt[3]{x0}\right)\]
   11. Using strategy rm

   12. Applied add-sqr-sqrt_binary64 2.6

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

   13. Applied add-sqr-sqrt_binary64 2.6

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

   14. Applied cbrt-prod_binary64 2.6

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

   15. Applied unswap-sqr_binary64 2.0

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

   16. Simplified 2.0

       \[\leadsto 1 \cdot \left(\left(\color{blue}{\left(\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}\right)} \cdot \left(\sqrt[3]{\sqrt{x0}} \cdot \sqrt{\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}}\right)\right) \cdot \sqrt[3]{x0}\right)\]

   17. Simplified 2.0

       \[\leadsto 1 \cdot \left(\left(\left(\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}\right) \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}\right)}\right) \cdot \sqrt[3]{x0}\right)\]
   18. Using strategy rm

   19. Applied flip--_binary64 2.3

       \[\leadsto 1 \cdot \left(\left(\left(\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}\right) \cdot \left(\sqrt{\color{blue}{\frac{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}}{\frac{\sqrt[3]{x0}}{1 - x1} + \sqrt[3]{x0}}}} \cdot {x0}^{0.16666666666666666}\right)\right) \cdot \sqrt[3]{x0}\right)\]

   20. Applied sqrt-div_binary64 2.3

       \[\leadsto 1 \cdot \left(\left(\left(\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}\right) \cdot \left(\color{blue}{\frac{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}}}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} + \sqrt[3]{x0}}}} \cdot {x0}^{0.16666666666666666}\right)\right) \cdot \sqrt[3]{x0}\right)\]

   21. Applied associate-*l/_binary64 2.3

       \[\leadsto 1 \cdot \left(\left(\left(\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}\right) \cdot \color{blue}{\frac{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} + \sqrt[3]{x0}}}}\right) \cdot \sqrt[3]{x0}\right)\]

   22. Applied flip3--_binary64 1.6

       \[\leadsto 1 \cdot \left(\left(\left(\sqrt{\color{blue}{\frac{{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}^{3} - {\left(\sqrt[3]{x0}\right)}^{3}}{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} + \left(\sqrt[3]{x0} \cdot \sqrt[3]{x0} + \frac{\sqrt[3]{x0}}{1 - x1} \cdot \sqrt[3]{x0}\right)}}} \cdot {x0}^{0.16666666666666666}\right) \cdot \frac{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} + \sqrt[3]{x0}}}\right) \cdot \sqrt[3]{x0}\right)\]

   23. Applied sqrt-div_binary64 1.6

       \[\leadsto 1 \cdot \left(\left(\left(\color{blue}{\frac{\sqrt{{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}^{3} - {\left(\sqrt[3]{x0}\right)}^{3}}}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} + \left(\sqrt[3]{x0} \cdot \sqrt[3]{x0} + \frac{\sqrt[3]{x0}}{1 - x1} \cdot \sqrt[3]{x0}\right)}}} \cdot {x0}^{0.16666666666666666}\right) \cdot \frac{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} + \sqrt[3]{x0}}}\right) \cdot \sqrt[3]{x0}\right)\]

   24. Applied associate-*l/_binary64 1.6

       \[\leadsto 1 \cdot \left(\left(\color{blue}{\frac{\sqrt{{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}^{3} - {\left(\sqrt[3]{x0}\right)}^{3}} \cdot {x0}^{0.16666666666666666}}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} + \left(\sqrt[3]{x0} \cdot \sqrt[3]{x0} + \frac{\sqrt[3]{x0}}{1 - x1} \cdot \sqrt[3]{x0}\right)}}} \cdot \frac{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} + \sqrt[3]{x0}}}\right) \cdot \sqrt[3]{x0}\right)\]

   25. Applied frac-times_binary64 1.6

       \[\leadsto 1 \cdot \left(\color{blue}{\frac{\left(\sqrt{{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}^{3} - {\left(\sqrt[3]{x0}\right)}^{3}} \cdot {x0}^{0.16666666666666666}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}} \cdot {x0}^{0.16666666666666666}\right)}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} + \left(\sqrt[3]{x0} \cdot \sqrt[3]{x0} + \frac{\sqrt[3]{x0}}{1 - x1} \cdot \sqrt[3]{x0}\right)} \cdot \sqrt{\frac{\sqrt[3]{x0}}{1 - x1} + \sqrt[3]{x0}}}} \cdot \sqrt[3]{x0}\right)\]

   if 0.98140000000000005 < (-.f64 1 x1)

   1. Initial program 11.3

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

   3. Applied *-un-lft-identity_binary64 11.3

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

   4. Applied *-un-lft-identity_binary64 11.3

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

   5. Applied distribute-lft-out--_binary64 11.3

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

   6. Simplified 10.5

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

   8. Applied add-cube-cbrt_binary64 10.5

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

   9. Applied associate-*r*_binary64 10.5

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

   10. Simplified 10.2

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

   11. Taylor expanded around 0 3.8

       \[\leadsto 1 \cdot \left(\left(\sqrt[3]{x0} \cdot \color{blue}{\left({x0}^{0.3333333333333333} \cdot x1 + \left({x0}^{0.3333333333333333} \cdot {x1}^{4} + \left({x0}^{0.3333333333333333} \cdot {x1}^{2} + {x0}^{0.3333333333333333} \cdot {x1}^{3}\right)\right)\right)}\right) \cdot \sqrt[3]{x0}\right)\]

   12. Simplified 3.8

       \[\leadsto 1 \cdot \left(\left(\sqrt[3]{x0} \cdot \color{blue}{\left(\sqrt[3]{x0} \cdot \left(x1 + \left({x1}^{3} + \left(x1 \cdot x1 + {x1}^{4}\right)\right)\right)\right)}\right) \cdot \sqrt[3]{x0}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x1 \leq 0.9814:\\ \;\;\;\;\sqrt[3]{x0} \cdot \frac{\left(\sqrt{{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}^{3} - {\left(\sqrt[3]{x0}\right)}^{3}} \cdot {x0}^{0.16666666666666666}\right) \cdot \left({x0}^{0.16666666666666666} \cdot \sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0} \cdot \sqrt[3]{x0}}\right)}{\sqrt{\frac{\sqrt[3]{x0}}{1 - x1} \cdot \frac{\sqrt[3]{x0}}{1 - x1} + \left(\sqrt[3]{x0} \cdot \sqrt[3]{x0} + \sqrt[3]{x0} \cdot \frac{\sqrt[3]{x0}}{1 - x1}\right)} \cdot \sqrt{\sqrt[3]{x0} + \frac{\sqrt[3]{x0}}{1 - x1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x0} \cdot \left(\sqrt[3]{x0} \cdot \left(\sqrt[3]{x0} \cdot \left(x1 + \left({x1}^{3} + \left(x1 \cdot x1 + {x1}^{4}\right)\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021126 
(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))