Logistic function from Lakshay Garg

Average Error: 29.1 → 0.0

Time: 1.8m

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.05005305256090602:\\ \;\;\;\;\frac{\frac{\frac{512}{\left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(\left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right)\right)} \cdot \frac{512}{\left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(\left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right)\right)} - 1}{\left(1 + \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)} \cdot \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right) + \left(\frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)} \cdot \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right) \cdot \left(\frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)} \cdot \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right)}}{\left(\left(\frac{2}{e^{-2 \cdot x} + 1} + 1\right) \cdot \frac{2}{e^{-2 \cdot x} + 1} + 1\right) \cdot \left(1 + \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right)}\\ \mathbf{elif}\;-2 \cdot x \le 0.0005413665455498676:\\ \;\;\;\;\left(x - \frac{1}{3} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{2}{15} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{e^{\frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)} \cdot \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}}}{e}\right)}{\left(\left(\frac{2}{e^{-2 \cdot x} + 1} + 1\right) \cdot \frac{2}{e^{-2 \cdot x} + 1} + 1\right) \cdot \left(1 + \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right)}\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r3093174 = 2.0;
        double r3093175 = 1.0;
        double r3093176 = -2.0;
        double r3093177 = x;
        double r3093178 = r3093176 * r3093177;
        double r3093179 = exp(r3093178);
        double r3093180 = r3093175 + r3093179;
        double r3093181 = r3093174 / r3093180;
        double r3093182 = r3093181 - r3093175;
        return r3093182;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r3093183 = -2.0;
        double r3093184 = x;
        double r3093185 = r3093183 * r3093184;
        double r3093186 = -0.05005305256090602;
        bool r3093187 = r3093185 <= r3093186;
        double r3093188 = 512.0;
        double r3093189 = exp(r3093185);
        double r3093190 = 1.0;
        double r3093191 = r3093189 + r3093190;
        double r3093192 = r3093191 * r3093191;
        double r3093193 = r3093192 * r3093191;
        double r3093194 = r3093193 * r3093193;
        double r3093195 = r3093193 * r3093194;
        double r3093196 = r3093188 / r3093195;
        double r3093197 = r3093196 * r3093196;
        double r3093198 = r3093197 - r3093190;
        double r3093199 = 8.0;
        double r3093200 = r3093199 / r3093193;
        double r3093201 = r3093200 * r3093200;
        double r3093202 = r3093190 + r3093201;
        double r3093203 = r3093201 * r3093201;
        double r3093204 = r3093202 + r3093203;
        double r3093205 = r3093198 / r3093204;
        double r3093206 = 2.0;
        double r3093207 = r3093206 / r3093191;
        double r3093208 = r3093207 + r3093190;
        double r3093209 = r3093208 * r3093207;
        double r3093210 = r3093209 + r3093190;
        double r3093211 = r3093190 + r3093200;
        double r3093212 = r3093210 * r3093211;
        double r3093213 = r3093205 / r3093212;
        double r3093214 = 0.0005413665455498676;
        bool r3093215 = r3093185 <= r3093214;
        double r3093216 = 0.3333333333333333;
        double r3093217 = r3093184 * r3093184;
        double r3093218 = r3093184 * r3093217;
        double r3093219 = r3093216 * r3093218;
        double r3093220 = r3093184 - r3093219;
        double r3093221 = 0.13333333333333333;
        double r3093222 = 5.0;
        double r3093223 = pow(r3093184, r3093222);
        double r3093224 = r3093221 * r3093223;
        double r3093225 = r3093220 + r3093224;
        double r3093226 = exp(r3093201);
        double r3093227 = exp(1.0);
        double r3093228 = r3093226 / r3093227;
        double r3093229 = log(r3093228);
        double r3093230 = r3093229 / r3093212;
        double r3093231 = r3093215 ? r3093225 : r3093230;
        double r3093232 = r3093187 ? r3093213 : r3093231;
        return r3093232;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 3 regimes

2. if (* -2 x) < -0.05005305256090602

   1. Initial program 0.0

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

   3. Applied flip3-- 0.0

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

   4. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} + \left(1 \cdot 1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 1\right)}\]

   5. Simplified 0.0

      \[\leadsto \frac{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1}{\color{blue}{1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}}}\]
   6. Using strategy rm

   7. Applied flip-- 0.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1 \cdot 1}{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1}}}{1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}}\]

   8. Applied associate-/l/ 0.0

      \[\leadsto \color{blue}{\frac{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1 \cdot 1}{\left(1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1\right)}}\]
   9. Using strategy rm

   10. Applied flip3-- 0.0

       \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}\right) \cdot \left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}\right) \cdot \left(1 \cdot 1\right)\right)}}}{\left(1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1\right)}\]

   11. Simplified 0.0

       \[\leadsto \frac{\frac{\color{blue}{\frac{512}{\left(\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right)\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right)\right)\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right)\right)} \cdot \frac{512}{\left(\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right)\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right)\right)\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right)\right)} - 1}}{\left(\frac{8}{\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{8}{\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)}\right) \cdot \left(\frac{8}{\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{8}{\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)}\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\frac{8}{\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{8}{\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)}\right) \cdot \left(1 \cdot 1\right)\right)}}{\left(1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{8}{\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\right)}\]

   if -0.05005305256090602 < (* -2 x) < 0.0005413665455498676

   1. Initial program 59.0

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

   3. Applied flip3-- 59.0

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

   4. Simplified 59.0

      \[\leadsto \frac{\color{blue}{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} + \left(1 \cdot 1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 1\right)}\]

   5. Simplified 59.0

      \[\leadsto \frac{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1}{\color{blue}{1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}}}\]

   6. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

   7. Simplified 0.0

      \[\leadsto \color{blue}{\frac{2}{15} \cdot {x}^{5} + \left(x - \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)}\]

   if 0.0005413665455498676 < (* -2 x)

   1. Initial program 0.0

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

   3. Applied flip3-- 0.0

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

   4. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1}}{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} + \left(1 \cdot 1 + \frac{2}{1 + e^{-2 \cdot x}} \cdot 1\right)}\]

   5. Simplified 0.0

      \[\leadsto \frac{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1}{\color{blue}{1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}}}\]
   6. Using strategy rm

   7. Applied flip-- 0.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1 \cdot 1}{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1}}}{1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}}\]

   8. Applied associate-/l/ 0.0

      \[\leadsto \color{blue}{\frac{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - 1 \cdot 1}{\left(1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1\right)}}\]
   9. Using strategy rm

   10. Applied add-log-exp 0.0

       \[\leadsto \frac{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - \color{blue}{\log \left(e^{1 \cdot 1}\right)}}{\left(1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1\right)}\]

   11. Applied add-log-exp 0.0

       \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}\right)} - \log \left(e^{1 \cdot 1}\right)}{\left(1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1\right)}\]

   12. Applied diff-log 0.0

       \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} \cdot \frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}}{e^{1 \cdot 1}}\right)}}{\left(1 + \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot \left(\frac{8}{\left(\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.05005305256090602:\\ \;\;\;\;\frac{\frac{\frac{512}{\left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(\left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right)\right)} \cdot \frac{512}{\left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(\left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right)\right)} - 1}{\left(1 + \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)} \cdot \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right) + \left(\frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)} \cdot \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right) \cdot \left(\frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)} \cdot \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right)}}{\left(\left(\frac{2}{e^{-2 \cdot x} + 1} + 1\right) \cdot \frac{2}{e^{-2 \cdot x} + 1} + 1\right) \cdot \left(1 + \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right)}\\ \mathbf{elif}\;-2 \cdot x \le 0.0005413665455498676:\\ \;\;\;\;\left(x - \frac{1}{3} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \frac{2}{15} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{e^{\frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)} \cdot \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}}}{e}\right)}{\left(\left(\frac{2}{e^{-2 \cdot x} + 1} + 1\right) \cdot \frac{2}{e^{-2 \cdot x} + 1} + 1\right) \cdot \left(1 + \frac{8}{\left(\left(e^{-2 \cdot x} + 1\right) \cdot \left(e^{-2 \cdot x} + 1\right)\right) \cdot \left(e^{-2 \cdot x} + 1\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))