Average Error: 8.4 → 2.9
Time: 4.4s
Precision: binary64
\[x0 = 1.855 \land x1 = 0.000209 \lor x0 = 2.985 \land x1 = 0.0186\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\begin{array}{l} \mathbf{if}\;1 - x1 \leq 0.9814:\\ \;\;\;\;\sqrt[3]{x0} \cdot \left(\sqrt[3]{\sqrt{x0}} \cdot \left(\sqrt[3]{\sqrt{x0}} \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x0 \cdot \left(x1 + \left({x1}^{4} + \left({x1}^{3} + {x1}^{2}\right)\right)\right)\\ \end{array}\]
\frac{x0}{1 - x1} - x0
↓
\begin{array}{l}
\mathbf{if}\;1 - x1 \leq 0.9814:\\
\;\;\;\;\sqrt[3]{x0} \cdot \left(\sqrt[3]{\sqrt{x0}} \cdot \left(\sqrt[3]{\sqrt{x0}} \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x0 \cdot \left(x1 + \left({x1}^{4} + \left({x1}^{3} + {x1}^{2}\right)\right)\right)\\

\end{array}
(FPCore (x0 x1) :precision binary64 (- (/ x0 (- 1.0 x1)) x0))
↓
(FPCore (x0 x1)
 :precision binary64
 (if (<= (- 1.0 x1) 0.9814)
   (*
    (cbrt x0)
    (*
     (cbrt (sqrt x0))
     (* (cbrt (sqrt x0)) (- (/ (cbrt x0) (- 1.0 x1)) (cbrt x0)))))
   (* x0 (+ x1 (+ (pow x1 4.0) (+ (pow x1 3.0) (pow x1 2.0)))))))
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) * (cbrt(sqrt(x0)) * (cbrt(sqrt(x0)) * ((cbrt(x0) / (1.0 - x1)) - cbrt(x0))));
	} else {
		tmp = x0 * (x1 + (pow(x1, 4.0) + (pow(x1, 3.0) + pow(x1, 2.0))));
	}
	return tmp;
}

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
Herbie2.9
\[\frac{x0 \cdot x1}{1 - x1}\]

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_51935.5

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

    4. Applied *-un-lft-identity_binary64_51935.5

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

    5. Applied distribute-lft-out--_binary64_51455.5

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

    6. Simplified4.4

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

    8. Applied add-cube-cbrt_binary64_52284.4

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

    9. Applied associate-*r*_binary64_51334.5

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

    10. Simplified2.6

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

    12. Applied add-sqr-sqrt_binary64_52152.6

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

    13. Applied cbrt-prod_binary64_52242.6

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

    14. Applied associate-*l*_binary64_51342.0

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

    15. Simplified2.0

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

    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_519311.3

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

    4. Applied *-un-lft-identity_binary64_519311.3

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

    5. Applied distribute-lft-out--_binary64_514511.3

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

    6. Simplified10.5

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

    7. Taylor expanded around 0 3.8

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x1 \leq 0.9814:\\ \;\;\;\;\sqrt[3]{x0} \cdot \left(\sqrt[3]{\sqrt{x0}} \cdot \left(\sqrt[3]{\sqrt{x0}} \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} - \sqrt[3]{x0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x0 \cdot \left(x1 + \left({x1}^{4} + \left({x1}^{3} + {x1}^{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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))