Average Error: 29.2 → 0.0
Time: 22.2s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0005525645036661:\\ \;\;\;\;\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\left(\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right)\right)\\ \mathbf{elif}\;x \le 0.001005004210539114:\\ \;\;\;\;x - \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\left(\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right)\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0005525645036661:\\
\;\;\;\;\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\left(\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right)\right)\\

\mathbf{elif}\;x \le 0.001005004210539114:\\
\;\;\;\;x - \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot x\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\left(\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right)\right)\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r1767075 = 2.0;
        double r1767076 = 1.0;
        double r1767077 = -2.0;
        double r1767078 = x;
        double r1767079 = r1767077 * r1767078;
        double r1767080 = exp(r1767079);
        double r1767081 = r1767076 + r1767080;
        double r1767082 = r1767075 / r1767081;
        double r1767083 = r1767082 - r1767076;
        return r1767083;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r1767084 = x;
        double r1767085 = -0.0005525645036661;
        bool r1767086 = r1767084 <= r1767085;
        double r1767087 = 2.0;
        double r1767088 = -2.0;
        double r1767089 = r1767088 * r1767084;
        double r1767090 = exp(r1767089);
        double r1767091 = 1.0;
        double r1767092 = r1767090 + r1767091;
        double r1767093 = r1767087 / r1767092;
        double r1767094 = sqrt(r1767093);
        double r1767095 = r1767094 - r1767091;
        double r1767096 = cbrt(r1767095);
        double r1767097 = r1767094 + r1767091;
        double r1767098 = r1767096 * r1767096;
        double r1767099 = cbrt(r1767098);
        double r1767100 = r1767096 * r1767099;
        double r1767101 = cbrt(r1767096);
        double r1767102 = r1767100 * r1767101;
        double r1767103 = r1767097 * r1767102;
        double r1767104 = r1767096 * r1767103;
        double r1767105 = 0.001005004210539114;
        bool r1767106 = r1767084 <= r1767105;
        double r1767107 = 0.3333333333333333;
        double r1767108 = r1767084 * r1767107;
        double r1767109 = r1767084 * r1767108;
        double r1767110 = r1767109 * r1767084;
        double r1767111 = r1767084 - r1767110;
        double r1767112 = r1767106 ? r1767111 : r1767104;
        double r1767113 = r1767086 ? r1767104 : r1767112;
        return r1767113;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0005525645036661 or 0.001005004210539114 < x

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied add-sqr-sqrt0.8

      \[\leadsto \color{blue}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}} - 1 \cdot 1\]

    5. Applied difference-of-squares0.6

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

    7. Applied add-cube-cbrt0.9

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

    8. Applied associate-*r*0.9

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

    10. Applied add-cube-cbrt0.9

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

    11. Applied cbrt-prod0.6

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

    12. Applied associate-*r*0.1

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

    if -0.0005525645036661 < x < 0.001005004210539114

    1. Initial program 59.2

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

    3. Applied *-un-lft-identity59.2

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

    4. Applied add-sqr-sqrt59.2

      \[\leadsto \color{blue}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}} - 1 \cdot 1\]

    5. Applied difference-of-squares59.2

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

    6. Taylor expanded around 0 60.2

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

    7. Simplified0.0

      \[\leadsto \color{blue}{\left(x + 0\right) - x \cdot \left(\left(\frac{1}{3} \cdot x\right) \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0005525645036661:\\ \;\;\;\;\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\left(\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right)\right)\\ \mathbf{elif}\;x \le 0.001005004210539114:\\ \;\;\;\;x - \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \left(\left(\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} + 1\right) \cdot \left(\left(\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1} \cdot \sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{2}{e^{-2 \cdot x} + 1}} - 1}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))