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

Average Error: 8.4 → 5.6

Time: 2.3s

Precision: 64

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

Math

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

↓

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

C

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

↓

double code(double x0, double x1) {
	double VAR;
	if (((1.0 - x1) <= 0.9905955)) {
		VAR = fma(sqrt((x0 / (1.0 - x1))), sqrt((x0 / (1.0 - x1))), -x0);
	} else {
		VAR = pow(pow(((double) M_E), (cbrt(log(fma(((cbrt(x0) * cbrt(x0)) / 1.0), (cbrt(x0) / (1.0 - x1)), -x0))) * cbrt(log(fma(((cbrt(x0) * cbrt(x0)) / 1.0), (cbrt(x0) / (1.0 - x1)), -x0))))), cbrt(log(fma(((cbrt(x0) * cbrt(x0)) / 1.0), (cbrt(x0) / (1.0 - x1)), -x0))));
	}
	return VAR;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	8.4
Target	0.5
Herbie	5.6

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

Derivation

1. Split input into 2 regimes

2. if (- 1.0 x1) < 0.9905955

   1. Initial program 5.5

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

   3. Applied add-sqr-sqrt 4.4

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

   4. Applied fma-neg 3.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{x0}{1 - x1}}, \sqrt{\frac{x0}{1 - x1}}, -x0\right)}\]

   if 0.9905955 < (- 1.0 x1)

   1. Initial program 11.3

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

   3. Applied *-un-lft-identity 11.3

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

   4. Applied add-cube-cbrt 11.3

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

   5. Applied times-frac 10.7

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

   6. Applied fma-neg 8.1

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

   8. Applied add-exp-log 8.1

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\right)}}\]
   9. Using strategy rm

   10. Applied pow1 8.1

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

   11. Applied log-pow 8.1

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

   12. Applied exp-prod 8.1

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

   13. Simplified 8.1

       \[\leadsto {\color{blue}{e}}^{\left(\log \left(\mathsf{fma}\left(\frac{\sqrt[3]{x0} \cdot \sqrt[3]{x0}}{1}, \frac{\sqrt[3]{x0}}{1 - x1}, -x0\right)\right)\right)}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 8.1

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

   16. Applied pow-unpow 8.0

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

4. Final simplification 5.6

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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))