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

Average Error: 7.9 → 2.1

Time: 11.8s

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

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

↓

\[\begin{array}{l} \mathbf{if}\;x1 \le 2.1826724243164061516238316773552696759 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

C

double f(double x0, double x1) {
        double r121433 = x0;
        double r121434 = 1.0;
        double r121435 = x1;
        double r121436 = r121434 - r121435;
        double r121437 = r121433 / r121436;
        double r121438 = r121437 - r121433;
        return r121438;
}

↓

double f(double x0, double x1) {
        double r121439 = x1;
        double r121440 = 0.00021826724243164062;
        bool r121441 = r121439 <= r121440;
        double r121442 = x0;
        double r121443 = 3.0;
        double r121444 = pow(r121442, r121443);
        double r121445 = 1.0;
        double r121446 = r121445 - r121439;
        double r121447 = 6.0;
        double r121448 = pow(r121446, r121447);
        double r121449 = r121444 / r121448;
        double r121450 = r121449 - r121444;
        double r121451 = exp(r121450);
        double r121452 = sqrt(r121451);
        double r121453 = log(r121452);
        double r121454 = r121453 + r121453;
        double r121455 = r121442 * r121442;
        double r121456 = r121446 * r121446;
        double r121457 = r121442 / r121456;
        double r121458 = r121457 + r121442;
        double r121459 = r121458 * r121457;
        double r121460 = r121455 + r121459;
        double r121461 = r121454 / r121460;
        double r121462 = r121442 * r121461;
        double r121463 = r121442 / r121446;
        double r121464 = r121463 + r121442;
        double r121465 = r121462 / r121464;
        double r121466 = sqrt(r121445);
        double r121467 = sqrt(r121439);
        double r121468 = r121466 + r121467;
        double r121469 = pow(r121468, r121447);
        double r121470 = r121466 - r121467;
        double r121471 = pow(r121470, r121447);
        double r121472 = r121469 * r121471;
        double r121473 = r121444 / r121472;
        double r121474 = r121473 - r121444;
        double r121475 = exp(r121474);
        double r121476 = log(r121475);
        double r121477 = r121476 / r121460;
        double r121478 = r121442 * r121477;
        double r121479 = r121478 / r121464;
        double r121480 = r121441 ? r121465 : r121479;
        return r121480;
}

Error

Bits error versus x0

Bits error versus x1

Target

Original	7.9
Target	0.2
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if x1 < 0.00021826724243164062

   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 0.00021826724243164062 < x1

   1. Initial program 4.5

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

   3. Applied flip-- 3.1

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

   4. Simplified 3.8

      \[\leadsto \frac{\color{blue}{x0 \cdot \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} - x0\right)}}{\frac{x0}{1 - x1} + x0}\]
   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 3.8

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

   14. Applied add-sqr-sqrt 3.8

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

   15. Applied add-sqr-sqrt 3.8

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

   16. Applied difference-of-squares 3.8

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x1 \le 2.1826724243164061516238316773552696759 \cdot 10^{-4}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right) + \log \left(\sqrt{e^{\frac{{x0}^{3}}{{\left(1 - x1\right)}^{6}} - {x0}^{3}}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x0 \cdot \frac{\log \left(e^{\frac{{x0}^{3}}{{\left(\sqrt{1} + \sqrt{x1}\right)}^{6} \cdot {\left(\sqrt{1} - \sqrt{x1}\right)}^{6}} - {x0}^{3}}\right)}{x0 \cdot x0 + \left(\frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)} + x0\right) \cdot \frac{x0}{\left(1 - x1\right) \cdot \left(1 - x1\right)}}}{\frac{x0}{1 - x1} + x0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :pre (or (and (== x0 1.855) (== x1 2.09000000000000012e-4)) (and (== x0 2.98499999999999988) (== x1 0.018599999999999998)))

  :herbie-target
  (/ (* x0 x1) (- 1 x1))

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