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

Average Error: 7.8 → 5.5

Time: 3.9s

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 r164011 = x0;
        double r164012 = 1.0;
        double r164013 = x1;
        double r164014 = r164012 - r164013;
        double r164015 = r164011 / r164014;
        double r164016 = r164015 - r164011;
        return r164016;
}

↓

double f(double x0, double x1) {
        double r164017 = 1.0;
        double r164018 = x0;
        double r164019 = exp(r164018);
        double r164020 = sqrt(r164019);
        double r164021 = r164017 / r164020;
        double r164022 = log(r164021);
        double r164023 = 3.0;
        double r164024 = pow(r164022, r164023);
        double r164025 = cbrt(r164018);
        double r164026 = 1.0;
        double r164027 = x1;
        double r164028 = r164026 - r164027;
        double r164029 = r164025 / r164028;
        double r164030 = 0.6666666666666666;
        double r164031 = pow(r164018, r164030);
        double r164032 = fma(r164029, r164031, r164022);
        double r164033 = pow(r164032, r164023);
        double r164034 = r164024 + r164033;
        double r164035 = r164029 * r164031;
        double r164036 = r164032 * r164035;
        double r164037 = log(r164020);
        double r164038 = r164037 * r164037;
        double r164039 = r164036 + r164038;
        double r164040 = r164034 / r164039;
        return r164040;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.8
Target	0.2
Herbie	5.5

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

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 6.9

   \[\leadsto \log \color{blue}{\left(\frac{{\left(e^{{x0}^{\frac{2}{3}}}\right)}^{\left(\frac{\sqrt[3]{x0}}{1 - x1}\right)}}{e^{x0}}\right)}\]
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 7.0

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

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 5.5

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

21. Final simplification 5.5

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

Reproduce

herbie shell --seed 2020024 +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))