Logistic function from Lakshay Garg

Average Error: 29.6 → 0.0

Time: 25.4s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.015791518006908589200065762270241975784 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(\frac{\left(\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}}{\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} \cdot \sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, \frac{\left(\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}}{\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} \cdot \sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}} + 1, \frac{\frac{\left(\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}}{{\left(\sqrt{1 + {\left(e^{x}\right)}^{-2}} \cdot \sqrt{1 + {\left(e^{x}\right)}^{-2}}\right)}^{\frac{1}{2}}}}{\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} \cdot \frac{\frac{{\left(\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}\right)}^{3}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}}}\right)} + \mathsf{fma}\left(-1, 1, 1\right)\\ \mathbf{elif}\;x \le 9.557681698529543615552861446360566333169 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(x, 1, -\mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, {x}^{3} \cdot 0.3333333333333333703407674875052180141211\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}} \cdot \frac{\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{\sqrt{{\left(e^{-2}\right)}^{x} + 1}}}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} - 1\right) + \mathsf{fma}\left(-1, 1, 1\right)\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r87268 = 2.0;
        double r87269 = 1.0;
        double r87270 = -2.0;
        double r87271 = x;
        double r87272 = r87270 * r87271;
        double r87273 = exp(r87272);
        double r87274 = r87269 + r87273;
        double r87275 = r87268 / r87274;
        double r87276 = r87275 - r87269;
        return r87276;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r87277 = x;
        double r87278 = -0.0007015791518006909;
        bool r87279 = r87277 <= r87278;
        double r87280 = 2.0;
        double r87281 = sqrt(r87280);
        double r87282 = cbrt(r87281);
        double r87283 = r87281 * r87282;
        double r87284 = r87282 * r87282;
        double r87285 = 1.0;
        double r87286 = exp(r87277);
        double r87287 = -2.0;
        double r87288 = pow(r87286, r87287);
        double r87289 = r87285 + r87288;
        double r87290 = sqrt(r87289);
        double r87291 = r87284 / r87290;
        double r87292 = r87283 * r87291;
        double r87293 = sqrt(r87290);
        double r87294 = r87293 * r87293;
        double r87295 = r87292 / r87294;
        double r87296 = 3.0;
        double r87297 = pow(r87295, r87296);
        double r87298 = pow(r87285, r87296);
        double r87299 = r87297 - r87298;
        double r87300 = r87295 + r87285;
        double r87301 = r87290 * r87290;
        double r87302 = 0.5;
        double r87303 = pow(r87301, r87302);
        double r87304 = r87292 / r87303;
        double r87305 = pow(r87293, r87296);
        double r87306 = r87305 / r87284;
        double r87307 = r87306 / r87283;
        double r87308 = r87293 * r87307;
        double r87309 = r87304 / r87308;
        double r87310 = fma(r87285, r87300, r87309);
        double r87311 = r87299 / r87310;
        double r87312 = -1.0;
        double r87313 = fma(r87312, r87285, r87285);
        double r87314 = r87311 + r87313;
        double r87315 = 0.0009557681698529544;
        bool r87316 = r87277 <= r87315;
        double r87317 = 5.551115123125783e-17;
        double r87318 = 4.0;
        double r87319 = pow(r87277, r87318);
        double r87320 = pow(r87277, r87296);
        double r87321 = 0.33333333333333337;
        double r87322 = r87320 * r87321;
        double r87323 = fma(r87317, r87319, r87322);
        double r87324 = -r87323;
        double r87325 = fma(r87277, r87285, r87324);
        double r87326 = r87284 / r87293;
        double r87327 = exp(r87287);
        double r87328 = pow(r87327, r87277);
        double r87329 = r87328 + r87285;
        double r87330 = sqrt(r87329);
        double r87331 = sqrt(r87330);
        double r87332 = r87283 / r87331;
        double r87333 = r87326 * r87332;
        double r87334 = r87333 / r87290;
        double r87335 = r87334 - r87285;
        double r87336 = r87335 + r87313;
        double r87337 = r87316 ? r87325 : r87336;
        double r87338 = r87279 ? r87314 : r87337;
        return r87338;
}

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 add-sqr-sqrt 0.1

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

   13. Applied sqrt-prod 0.1

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

   14. Applied add-cube-cbrt 0.1

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

   15. Applied times-frac 0.1

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

   16. Applied associate-*l* 0.1

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

   17. Simplified 0.1

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

   19. Applied flip3-- 0.1

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

   20. Simplified 0.1

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

   21. Simplified 0.1

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

   23. Applied pow1/2 0.1

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

   24. Applied pow1/2 0.1

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

   25. Applied pow-prod-down 0.1

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

   26. Simplified 0.1

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

   if -0.0007015791518006909 < x < 0.0009557681698529544

   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 0.0009557681698529544 < 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 add-sqr-sqrt 1.6

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

   13. Applied sqrt-prod 1.6

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

   14. Applied add-cube-cbrt 1.6

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

   15. Applied times-frac 1.6

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

   16. Applied associate-*l* 0.1

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

   17. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.015791518006908589200065762270241975784 \cdot 10^{-4}:\\ \;\;\;\;\frac{{\left(\frac{\left(\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}}{\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} \cdot \sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}}\right)}^{3} - {1}^{3}}{\mathsf{fma}\left(1, \frac{\left(\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}}{\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} \cdot \sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}} + 1, \frac{\frac{\left(\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}}{{\left(\sqrt{1 + {\left(e^{x}\right)}^{-2}} \cdot \sqrt{1 + {\left(e^{x}\right)}^{-2}}\right)}^{\frac{1}{2}}}}{\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} \cdot \frac{\frac{{\left(\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}\right)}^{3}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}}{\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}}}\right)} + \mathsf{fma}\left(-1, 1, 1\right)\\ \mathbf{elif}\;x \le 9.557681698529543615552861446360566333169 \cdot 10^{-4}:\\ \;\;\;\;\mathsf{fma}\left(x, 1, -\mathsf{fma}\left(5.5511151231257827021181583404541015625 \cdot 10^{-17}, {x}^{4}, {x}^{3} \cdot 0.3333333333333333703407674875052180141211\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{\sqrt{1 + {\left(e^{x}\right)}^{-2}}}} \cdot \frac{\sqrt{2} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{\sqrt{{\left(e^{-2}\right)}^{x} + 1}}}}{\sqrt{1 + {\left(e^{x}\right)}^{-2}}} - 1\right) + \mathsf{fma}\left(-1, 1, 1\right)\\ \end{array}\]

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))