Logistic function from Lakshay Garg

Average Error: 29.6 → 0.0

Time: 28.9s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.015791518006908589200065762270241975784 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(-1, 1, 1\right) + \left(\sqrt{\frac{2}{\frac{{\left(e^{-2}\right)}^{x} + 1}{\frac{2}{{\left(e^{-2}\right)}^{x} + 1}}}} - 1\right)\\ \mathbf{elif}\;x \le 8.820524827028167679995873662335270637413 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(x, 1, -\mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\mathsf{fma}\left(\sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\frac{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}{\sqrt{2}}}}, \sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}} \cdot \frac{\sqrt{2}}{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\frac{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}{\sqrt{2}}}}, \sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}} \cdot \frac{\sqrt{2}}{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}}, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\frac{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}{\sqrt{2}}}}, \sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}} \cdot \frac{\sqrt{2}}{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}}, -1\right)} + \mathsf{fma}\left(1, -1, 1\right)\right) + \mathsf{fma}\left(-1, 1, 1\right)\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r87326 = 2.0;
        double r87327 = 1.0;
        double r87328 = -2.0;
        double r87329 = x;
        double r87330 = r87328 * r87329;
        double r87331 = exp(r87330);
        double r87332 = r87327 + r87331;
        double r87333 = r87326 / r87332;
        double r87334 = r87333 - r87327;
        return r87334;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r87335 = x;
        double r87336 = -0.0007015791518006909;
        bool r87337 = r87335 <= r87336;
        double r87338 = -1.0;
        double r87339 = 1.0;
        double r87340 = fma(r87338, r87339, r87339);
        double r87341 = 2.0;
        double r87342 = -2.0;
        double r87343 = exp(r87342);
        double r87344 = pow(r87343, r87335);
        double r87345 = r87344 + r87339;
        double r87346 = r87341 / r87345;
        double r87347 = r87345 / r87346;
        double r87348 = r87341 / r87347;
        double r87349 = sqrt(r87348);
        double r87350 = r87349 - r87339;
        double r87351 = r87340 + r87350;
        double r87352 = 0.0008820524827028168;
        bool r87353 = r87335 <= r87352;
        double r87354 = 5.551115123125783e-17;
        double r87355 = 4.0;
        double r87356 = pow(r87335, r87355);
        double r87357 = 0.33333333333333337;
        double r87358 = 3.0;
        double r87359 = pow(r87335, r87358);
        double r87360 = r87357 * r87359;
        double r87361 = fma(r87354, r87356, r87360);
        double r87362 = -r87361;
        double r87363 = fma(r87335, r87339, r87362);
        double r87364 = sqrt(r87341);
        double r87365 = exp(r87335);
        double r87366 = pow(r87365, r87342);
        double r87367 = r87366 + r87339;
        double r87368 = cbrt(r87367);
        double r87369 = r87364 / r87368;
        double r87370 = sqrt(r87367);
        double r87371 = r87370 / r87364;
        double r87372 = r87369 / r87371;
        double r87373 = sqrt(r87372);
        double r87374 = r87369 / r87368;
        double r87375 = r87364 / r87370;
        double r87376 = r87374 * r87375;
        double r87377 = sqrt(r87376);
        double r87378 = -r87339;
        double r87379 = fma(r87373, r87377, r87378);
        double r87380 = cbrt(r87379);
        double r87381 = r87380 * r87380;
        double r87382 = r87381 * r87380;
        double r87383 = fma(r87339, r87338, r87339);
        double r87384 = r87382 + r87383;
        double r87385 = r87384 + r87340;
        double r87386 = r87353 ? r87363 : r87385;
        double r87387 = r87337 ? r87351 : r87386;
        return r87387;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if x < -0.0007015791518006909

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   4. Applied add-cube-cbrt 0.0

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

   5. Applied add-sqr-sqrt 0.1

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

   6. Applied add-sqr-sqrt 0.1

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

   7. Applied times-frac 0.1

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

   8. Applied prod-diff 0.1

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

   9. Simplified 0.1

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

   10. Simplified 0.1

       \[\leadsto \left(\frac{\frac{\sqrt{2}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} \cdot \sqrt{2}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} - 1\right) + \color{blue}{\mathsf{fma}\left(-1, 1, 1\right)}\]
   11. Using strategy rm

   12. Applied sqrt-undiv 0.1

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

   13. Applied sqrt-unprod 0.0

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

   14. Applied sqrt-undiv 0.0

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

   15. Simplified 0.0

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

   if -0.0007015791518006909 < x < 0.0008820524827028168

   1. Initial program 59.2

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

   2. Simplified 59.2

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

   3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{1 \cdot x - \left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4}\right)}\]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1, -\mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, {x}^{3} \cdot 0.3333333333333333703407674875052180141211\right)\right)}\]

   if 0.0008820524827028168 < x

   1. Initial program 0.0

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

   2. Simplified 0.0

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

   4. Applied add-cube-cbrt 0.0

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

   5. Applied add-sqr-sqrt 0.1

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

   6. Applied add-sqr-sqrt 1.6

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

   7. Applied times-frac 1.6

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

   8. Applied prod-diff 1.0

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

   9. Simplified 1.6

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

   10. Simplified 1.6

       \[\leadsto \left(\frac{\frac{\sqrt{2}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} \cdot \sqrt{2}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} - 1\right) + \color{blue}{\mathsf{fma}\left(-1, 1, 1\right)}\]
   11. Using strategy rm

   12. Applied sqrt-undiv 1.6

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

   13. Applied sqrt-unprod 0.1

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

   14. Applied sqrt-undiv 0.0

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

   15. Simplified 0.0

       \[\leadsto \left(\sqrt{\color{blue}{\frac{2}{\frac{1 + {\left(e^{-2}\right)}^{x}}{\frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}}} - 1\right) + \mathsf{fma}\left(-1, 1, 1\right)\]
   16. Using strategy rm

   17. Applied add-sqr-sqrt 0.0

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

   18. Applied add-cube-cbrt 0.1

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

   19. Applied add-sqr-sqrt 0.1

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

   20. Applied times-frac 0.1

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

   21. Applied add-sqr-sqrt 0.1

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

   22. Applied times-frac 0.1

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

   23. Applied add-sqr-sqrt 1.6

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

   24. Applied times-frac 1.6

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

   25. Applied sqrt-prod 1.6

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

   26. Applied prod-diff 1.0

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

   27. Simplified 1.0

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

   28. Simplified 1.0

       \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{\sqrt{2}}{\frac{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}}}, \sqrt{\frac{\sqrt{2}}{\frac{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}}}, -1\right) + \color{blue}{\mathsf{fma}\left(1, -1, 1\right)}\right) + \mathsf{fma}\left(-1, 1, 1\right)\]
   29. Using strategy rm

   30. Applied add-cube-cbrt 0.1

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

   31. Simplified 0.1

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

   32. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.015791518006908589200065762270241975784 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(-1, 1, 1\right) + \left(\sqrt{\frac{2}{\frac{{\left(e^{-2}\right)}^{x} + 1}{\frac{2}{{\left(e^{-2}\right)}^{x} + 1}}}} - 1\right)\\ \mathbf{elif}\;x \le 8.820524827028167679995873662335270637413 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(x, 1, -\mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\mathsf{fma}\left(\sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\frac{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}{\sqrt{2}}}}, \sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}} \cdot \frac{\sqrt{2}}{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}}, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\frac{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}{\sqrt{2}}}}, \sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}} \cdot \frac{\sqrt{2}}{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}}, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\frac{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}{\sqrt{2}}}}, \sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}}}{\sqrt[3]{{\left(e^{x}\right)}^{-2} + 1}} \cdot \frac{\sqrt{2}}{\sqrt{{\left(e^{x}\right)}^{-2} + 1}}}, -1\right)} + \mathsf{fma}\left(1, -1, 1\right)\right) + \mathsf{fma}\left(-1, 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))