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

Average Error: 7.8 → 5.4

Time: 4.0s

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

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

↓

\[\frac{{\left(\log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)}^{3} + {\left(\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right) \cdot \left(\frac{\sqrt[3]{x0}}{1 - x1} \cdot {x0}^{\frac{2}{3}}\right) + \log \left(\sqrt{e^{x0}}\right) \cdot \log \left(\sqrt{e^{x0}}\right)}\]

C

double f(double x0, double x1) {
        double r162581 = x0;
        double r162582 = 1.0;
        double r162583 = x1;
        double r162584 = r162582 - r162583;
        double r162585 = r162581 / r162584;
        double r162586 = r162585 - r162581;
        return r162586;
}

↓

double f(double x0, double x1) {
        double r162587 = 1.0;
        double r162588 = x0;
        double r162589 = exp(r162588);
        double r162590 = sqrt(r162589);
        double r162591 = r162587 / r162590;
        double r162592 = log(r162591);
        double r162593 = 3.0;
        double r162594 = pow(r162592, r162593);
        double r162595 = cbrt(r162588);
        double r162596 = 1.0;
        double r162597 = x1;
        double r162598 = r162596 - r162597;
        double r162599 = r162595 / r162598;
        double r162600 = 0.6666666666666666;
        double r162601 = pow(r162588, r162600);
        double r162602 = fma(r162599, r162601, r162592);
        double r162603 = pow(r162602, r162593);
        double r162604 = r162594 + r162603;
        double r162605 = r162599 * r162601;
        double r162606 = r162602 * r162605;
        double r162607 = log(r162590);
        double r162608 = r162607 * r162607;
        double r162609 = r162606 + r162608;
        double r162610 = r162604 / r162609;
        return r162610;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.3
Herbie	5.4

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

Derivation

1. Initial program 7.8

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

3. Applied *-un-lft-identity 7.8

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

4. Applied add-cube-cbrt 7.8

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

5. Applied times-frac 8.2

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

6. Applied fma-neg 6.9

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

8. Applied add-log-exp 7.8

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

9. Simplified 6.8

   \[\leadsto \log \color{blue}{\left(\frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{e^{x0}}\right)}\]
10. Using strategy rm

11. Applied add-sqr-sqrt 7.3

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

12. Applied *-un-lft-identity 7.3

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

13. Applied unpow-prod-down 7.3

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

14. Applied times-frac 6.9

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

15. Applied log-prod 7.0

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

16. Simplified 7.0

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

17. Simplified 6.4

    \[\leadsto \log \left(\frac{1}{\sqrt{e^{x0}}}\right) + \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x0}}{1 - x1}, {x0}^{\frac{2}{3}}, \log \left(\frac{1}{\sqrt{e^{x0}}}\right)\right)}\]
18. Using strategy rm

19. Applied flip3-+ 5.5

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

20. Simplified 5.4

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

21. Final simplification 5.4

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

Reproduce

herbie shell --seed 2020062 +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))