Logistic function from Lakshay Garg

Average Error: 29.5 → 0.1

Time: 4.6s

Precision: 64

• unused variable y

Math

\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.1154491279793497:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\left({2}^{3} \cdot 2\right) \cdot {\left(\frac{\sqrt[3]{2}}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.7050588330471929 \cdot 10^{-4}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}^{3}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r59438 = 2.0;
        double r59439 = 1.0;
        double r59440 = -2.0;
        double r59441 = x;
        double r59442 = r59440 * r59441;
        double r59443 = exp(r59442);
        double r59444 = r59439 + r59443;
        double r59445 = r59438 / r59444;
        double r59446 = r59445 - r59439;
        return r59446;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r59447 = -2.0;
        double r59448 = x;
        double r59449 = r59447 * r59448;
        double r59450 = -1.1154491279793497;
        bool r59451 = r59449 <= r59450;
        double r59452 = 2.0;
        double r59453 = 3.0;
        double r59454 = pow(r59452, r59453);
        double r59455 = r59454 * r59452;
        double r59456 = cbrt(r59452);
        double r59457 = 1.0;
        double r59458 = exp(r59449);
        double r59459 = r59457 + r59458;
        double r59460 = r59456 / r59459;
        double r59461 = 6.0;
        double r59462 = pow(r59460, r59461);
        double r59463 = r59455 * r59462;
        double r59464 = sqrt(r59463);
        double r59465 = cbrt(r59464);
        double r59466 = r59465 + r59457;
        double r59467 = r59452 / r59459;
        double r59468 = pow(r59467, r59461);
        double r59469 = sqrt(r59468);
        double r59470 = cbrt(r59469);
        double r59471 = r59470 - r59457;
        double r59472 = r59466 * r59471;
        double r59473 = r59467 + r59457;
        double r59474 = r59472 / r59473;
        double r59475 = 0.0001705058833047193;
        bool r59476 = r59449 <= r59475;
        double r59477 = r59457 * r59448;
        double r59478 = 5.551115123125783e-17;
        double r59479 = 4.0;
        double r59480 = pow(r59448, r59479);
        double r59481 = r59478 * r59480;
        double r59482 = 0.33333333333333337;
        double r59483 = pow(r59448, r59453);
        double r59484 = r59482 * r59483;
        double r59485 = r59481 + r59484;
        double r59486 = r59477 - r59485;
        double r59487 = r59470 + r59457;
        double r59488 = pow(r59471, r59453);
        double r59489 = cbrt(r59488);
        double r59490 = r59487 * r59489;
        double r59491 = r59490 / r59473;
        double r59492 = r59476 ? r59486 : r59491;
        double r59493 = r59451 ? r59474 : r59492;
        return r59493;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if (* -2.0 x) < -1.1154491279793497

   1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
   2. Using strategy rm

   3. Applied flip-- 0.0

      \[\leadsto \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 0.0

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{\color{blue}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   6. Applied add-cbrt-cube 0.0

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   7. Applied cbrt-undiv 0.0

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   8. Applied add-cbrt-cube 0.0

      \[\leadsto \frac{\frac{2}{\color{blue}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} \cdot \sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   9. Applied add-cbrt-cube 0.0

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} \cdot \sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   10. Applied cbrt-undiv 0.0

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} \cdot \sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   11. Applied cbrt-unprod 0.0

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   12. Simplified 0.0

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 0.0

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}} \cdot \sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   15. Applied cbrt-prod 0.0

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   16. Applied difference-of-squares 0.0

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
   17. Using strategy rm

   18. Applied *-un-lft-identity 0.0

       \[\leadsto \frac{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{\color{blue}{1 \cdot \left(1 + e^{-2 \cdot x}\right)}}\right)}^{6}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   19. Applied add-cube-cbrt 1.6

       \[\leadsto \frac{\left(\sqrt[3]{\sqrt{{\left(\frac{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{1 \cdot \left(1 + e^{-2 \cdot x}\right)}\right)}^{6}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   20. Applied times-frac 1.6

       \[\leadsto \frac{\left(\sqrt[3]{\sqrt{{\color{blue}{\left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1} \cdot \frac{\sqrt[3]{2}}{1 + e^{-2 \cdot x}}\right)}}^{6}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   21. Applied unpow-prod-down 1.6

       \[\leadsto \frac{\left(\sqrt[3]{\sqrt{\color{blue}{{\left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{1}\right)}^{6} \cdot {\left(\frac{\sqrt[3]{2}}{1 + e^{-2 \cdot x}}\right)}^{6}}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   22. Simplified 0.0

       \[\leadsto \frac{\left(\sqrt[3]{\sqrt{\color{blue}{\left({2}^{3} \cdot 2\right)} \cdot {\left(\frac{\sqrt[3]{2}}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   if -1.1154491279793497 < (* -2.0 x) < 0.0001705058833047193

   1. Initial program 59.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

   2. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

   if 0.0001705058833047193 < (* -2.0 x)

   1. Initial program 0.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
   2. Using strategy rm

   3. Applied flip-- 0.1

      \[\leadsto \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube 0.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{\color{blue}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   6. Applied add-cbrt-cube 0.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   7. Applied cbrt-undiv 0.1

      \[\leadsto \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   8. Applied add-cbrt-cube 0.1

      \[\leadsto \frac{\frac{2}{\color{blue}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} \cdot \sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   9. Applied add-cbrt-cube 0.1

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} \cdot \sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   10. Applied cbrt-undiv 0.1

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} \cdot \sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   11. Applied cbrt-unprod 0.1

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   12. Simplified 0.1

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 0.1

       \[\leadsto \frac{\sqrt[3]{\color{blue}{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}} \cdot \sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   15. Applied cbrt-prod 0.1

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} \cdot \sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   16. Applied difference-of-squares 0.1

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]
   17. Using strategy rm

   18. Applied add-cbrt-cube 0.1

       \[\leadsto \frac{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\]

   19. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.1154491279793497:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\left({2}^{3} \cdot 2\right) \cdot {\left(\frac{\sqrt[3]{2}}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.7050588330471929 \cdot 10^{-4}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} + 1\right) \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{6}}} - 1\right)}^{3}}}{\frac{2}{1 + e^{-2 \cdot x}} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))