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

Average Error: 7.9 → 2.1

Time: 8.8s

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 6.625985107421873947924906289586033381056 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r208568 = x0;
        double r208569 = 1.0;
        double r208570 = x1;
        double r208571 = r208569 - r208570;
        double r208572 = r208568 / r208571;
        double r208573 = r208572 - r208568;
        return r208573;
}

↓

double f(double x0, double x1) {
        double r208574 = x1;
        double r208575 = 0.0006625985107421874;
        bool r208576 = r208574 <= r208575;
        double r208577 = x0;
        double r208578 = 3.0;
        double r208579 = pow(r208577, r208578);
        double r208580 = 1.0;
        double r208581 = r208580 - r208574;
        double r208582 = 6.0;
        double r208583 = pow(r208581, r208582);
        double r208584 = r208579 / r208583;
        double r208585 = r208584 - r208579;
        double r208586 = exp(r208585);
        double r208587 = sqrt(r208586);
        double r208588 = log(r208587);
        double r208589 = r208588 + r208588;
        double r208590 = r208577 * r208577;
        double r208591 = r208581 * r208581;
        double r208592 = r208577 / r208591;
        double r208593 = r208592 + r208577;
        double r208594 = r208593 * r208592;
        double r208595 = r208590 + r208594;
        double r208596 = r208589 / r208595;
        double r208597 = r208577 * r208596;
        double r208598 = r208577 / r208581;
        double r208599 = r208598 + r208577;
        double r208600 = r208597 / r208599;
        double r208601 = sqrt(r208580);
        double r208602 = sqrt(r208574);
        double r208603 = r208601 + r208602;
        double r208604 = pow(r208603, r208582);
        double r208605 = r208601 - r208602;
        double r208606 = pow(r208605, r208582);
        double r208607 = r208604 * r208606;
        double r208608 = r208579 / r208607;
        double r208609 = r208608 - r208579;
        double r208610 = exp(r208609);
        double r208611 = log(r208610);
        double r208612 = r208611 / r208595;
        double r208613 = r208577 * r208612;
        double r208614 = r208613 / r208599;
        double r208615 = r208576 ? r208600 : r208614;
        return r208615;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.3
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if x1 < 0.0006625985107421874

   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 2.6

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

   if 0.0006625985107421874 < 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 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 6.625985107421873947924906289586033381056 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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))