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

Average Error: 7.9 → 2.4

Time: 8.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program pre

  1. x0 = 2.985
  2. x1 = 0.000209

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \frac{x0}{1 - x1} \cdot \left(\frac{x0}{{\left(1 - x1\right)}^{3}} + \frac{x0}{1 - x1}\right)}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \frac{x0}{1 - x1} \cdot \left(\frac{x0}{{\left(1 - x1\right)}^{3}} + \frac{x0}{1 - x1}\right)}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r183712 = x0;
        double r183713 = 1.0;
        double r183714 = x1;
        double r183715 = r183713 - r183714;
        double r183716 = r183712 / r183715;
        double r183717 = r183716 - r183712;
        return r183717;
}

↓

double f(double x0, double x1) {
        double r183718 = x1;
        double r183719 = 0.00021208908081054686;
        bool r183720 = r183718 <= r183719;
        double r183721 = x0;
        double r183722 = 3.0;
        double r183723 = pow(r183721, r183722);
        double r183724 = 1.0;
        double r183725 = r183724 - r183718;
        double r183726 = 6.0;
        double r183727 = pow(r183725, r183726);
        double r183728 = r183723 / r183727;
        double r183729 = r183728 - r183723;
        double r183730 = exp(r183729);
        double r183731 = sqrt(r183730);
        double r183732 = log(r183731);
        double r183733 = r183732 + r183732;
        double r183734 = r183721 * r183721;
        double r183735 = r183721 / r183725;
        double r183736 = pow(r183725, r183722);
        double r183737 = r183721 / r183736;
        double r183738 = r183737 + r183735;
        double r183739 = r183735 * r183738;
        double r183740 = r183734 + r183739;
        double r183741 = r183733 / r183740;
        double r183742 = r183721 * r183741;
        double r183743 = r183735 + r183721;
        double r183744 = r183742 / r183743;
        double r183745 = sqrt(r183724);
        double r183746 = sqrt(r183718);
        double r183747 = r183745 + r183746;
        double r183748 = pow(r183747, r183726);
        double r183749 = r183745 - r183746;
        double r183750 = pow(r183749, r183726);
        double r183751 = r183748 * r183750;
        double r183752 = r183723 / r183751;
        double r183753 = r183752 - r183723;
        double r183754 = exp(r183753);
        double r183755 = log(r183754);
        double r183756 = r183755 / r183740;
        double r183757 = r183721 * r183756;
        double r183758 = r183757 / r183743;
        double r183759 = r183720 ? r183744 : r183758;
        return r183759;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.3
Herbie	2.4

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

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 5.9

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

   9. Applied add-log-exp 5.9

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

   10. Applied add-log-exp 5.9

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

   11. Applied diff-log 5.0

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

   12. Simplified 5.0

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

   14. Applied add-sqr-sqrt 3.4

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

   15. Applied log-prod 3.0

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

   if 0.00021208908081054686 < x1

   1. Initial program 4.5

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

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

   9. Applied add-log-exp 3.9

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

   10. Applied add-log-exp 3.9

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

   11. Applied diff-log 3.9

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

   12. Simplified 3.8

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

   14. Applied add-sqr-sqrt 3.8

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

   15. Applied add-sqr-sqrt 3.8

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

   16. Applied difference-of-squares 3.8

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

   17. Applied unpow-prod-down 1.8

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.12089080810546861 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \frac{x0}{1 - x1} \cdot \left(\frac{x0}{{\left(1 - x1\right)}^{3}} + \frac{x0}{1 - x1}\right)}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \frac{x0}{1 - x1} \cdot \left(\frac{x0}{{\left(1 - x1\right)}^{3}} + \frac{x0}{1 - x1}\right)}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))