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

Average Error: 7.9 → 2.0

Time: 12.5s

Precision: 64

• could not determine a ground truth for program pre

  1. x0 = 1.855
  2. x1 = 0.0186

\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861321705391922876060562 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r122881 = x0;
        double r122882 = 1.0;
        double r122883 = x1;
        double r122884 = r122882 - r122883;
        double r122885 = r122881 / r122884;
        double r122886 = r122885 - r122881;
        return r122886;
}

↓

double f(double x0, double x1) {
        double r122887 = x1;
        double r122888 = 0.00021208908081054686;
        bool r122889 = r122887 <= r122888;
        double r122890 = x0;
        double r122891 = 3.0;
        double r122892 = pow(r122890, r122891);
        double r122893 = 1.0;
        double r122894 = r122893 - r122887;
        double r122895 = 6.0;
        double r122896 = pow(r122894, r122895);
        double r122897 = r122892 / r122896;
        double r122898 = r122897 - r122892;
        double r122899 = exp(r122898);
        double r122900 = sqrt(r122899);
        double r122901 = log(r122900);
        double r122902 = r122901 + r122901;
        double r122903 = r122890 * r122890;
        double r122904 = r122894 * r122894;
        double r122905 = r122890 / r122904;
        double r122906 = r122905 + r122890;
        double r122907 = r122906 * r122905;
        double r122908 = r122903 + r122907;
        double r122909 = r122902 / r122908;
        double r122910 = r122890 * r122909;
        double r122911 = r122890 / r122894;
        double r122912 = r122911 + r122890;
        double r122913 = r122910 / r122912;
        double r122914 = sqrt(r122893);
        double r122915 = sqrt(r122887);
        double r122916 = r122914 + r122915;
        double r122917 = pow(r122916, r122895);
        double r122918 = r122914 - r122915;
        double r122919 = pow(r122918, r122895);
        double r122920 = r122917 * r122919;
        double r122921 = r122892 / r122920;
        double r122922 = r122921 - r122892;
        double r122923 = exp(r122922);
        double r122924 = log(r122923);
        double r122925 = r122924 / r122908;
        double r122926 = r122890 * r122925;
        double r122927 = r122926 / r122912;
        double r122928 = r122889 ? r122913 : r122927;
        return r122928;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.3
Herbie	2.0

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

Derivation

1. Split input into 2 regimes

2. if x1 < 0.00021208908081054686

   1. Initial program 11.2

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

   3. Applied flip-- 11.4

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

   4. Simplified 8.7

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

   6. Applied flip3-- 6.0

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

   7. Simplified 6.0

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

   9. Applied add-log-exp 6.0

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

   10. Applied add-log-exp 6.0

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

   11. Applied diff-log 5.2

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

   12. Simplified 5.2

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 3.5

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}} \cdot \sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   15. Applied log-prod 2.5

       \[\leadsto \frac{x0 \cdot \frac{\color{blue}{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   if 0.00021208908081054686 < x1

   1. Initial program 4.6

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

   3. Applied flip-- 3.2

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

   4. Simplified 3.9

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

   6. Applied flip3-- 3.9

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

   7. Simplified 3.9

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

   9. Applied add-log-exp 3.9

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

   10. Applied add-log-exp 3.9

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

   11. Applied diff-log 4.0

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

   12. Simplified 3.9

       \[\leadsto \frac{x0 \cdot \frac{\log \color{blue}{\left(e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}\right)}}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 3.9

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(1 - \color{blue}{\sqrt{x1} \cdot \sqrt{x1}}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   15. Applied add-sqr-sqrt 3.9

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \sqrt{x1} \cdot \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   16. Applied difference-of-squares 3.9

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\color{blue}{\left(\left(\sqrt{1} + \sqrt{x1}\right) \cdot \left(\sqrt{1} - \sqrt{x1}\right)\right)}}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]

   17. Applied unpow-prod-down 1.6

       \[\leadsto \frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{\color{blue}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861321705391922876060562 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))