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

Average Error: 7.9 → 2.4

Time: 7.8s

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

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

Math

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

↓

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

C

double f(double x0, double x1) {
        double r161464 = x0;
        double r161465 = 1.0;
        double r161466 = x1;
        double r161467 = r161465 - r161466;
        double r161468 = r161464 / r161467;
        double r161469 = r161468 - r161464;
        return r161469;
}

↓

double f(double x0, double x1) {
        double r161470 = x1;
        double r161471 = 0.00021208908081054686;
        bool r161472 = r161470 <= r161471;
        double r161473 = x0;
        double r161474 = 3.0;
        double r161475 = pow(r161473, r161474);
        double r161476 = 1.0;
        double r161477 = r161476 - r161470;
        double r161478 = 6.0;
        double r161479 = pow(r161477, r161478);
        double r161480 = r161475 / r161479;
        double r161481 = r161480 - r161475;
        double r161482 = exp(r161481);
        double r161483 = sqrt(r161482);
        double r161484 = log(r161483);
        double r161485 = r161484 + r161484;
        double r161486 = r161473 / r161477;
        double r161487 = pow(r161477, r161474);
        double r161488 = r161473 / r161487;
        double r161489 = r161486 + r161488;
        double r161490 = r161486 * r161489;
        double r161491 = r161473 * r161473;
        double r161492 = r161490 + r161491;
        double r161493 = r161485 / r161492;
        double r161494 = r161473 * r161493;
        double r161495 = r161486 + r161473;
        double r161496 = r161494 / r161495;
        double r161497 = sqrt(r161476);
        double r161498 = sqrt(r161470);
        double r161499 = r161497 + r161498;
        double r161500 = pow(r161499, r161478);
        double r161501 = r161497 - r161498;
        double r161502 = pow(r161501, r161478);
        double r161503 = r161500 * r161502;
        double r161504 = r161475 / r161503;
        double r161505 = r161504 - r161475;
        double r161506 = exp(r161505);
        double r161507 = log(r161506);
        double r161508 = r161507 / r161492;
        double r161509 = r161473 * r161508;
        double r161510 = r161509 / r161495;
        double r161511 = r161472 ? r161496 : r161510;
        return r161511;
}

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

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

   9. Applied add-log-exp 5.9

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

   10. Applied add-log-exp 5.9

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

   11. Applied diff-log 5.1

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

   12. Simplified 5.1

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

   14. Applied add-sqr-sqrt 3.4

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

   15. Applied log-prod 3.0

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

   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

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

   9. Applied add-log-exp 3.9

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

   10. Applied add-log-exp 3.9

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

   11. Applied diff-log 3.9

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

   12. Simplified 3.8

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

   14. Applied add-sqr-sqrt 3.8

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

   15. Applied add-sqr-sqrt 3.8

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

   16. Applied difference-of-squares 3.8

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

   17. Applied unpow-prod-down 1.8

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

4. Final simplification 2.4

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

Reproduce

herbie shell --seed 2020046 
(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))