Average Error: 29.0 → 0.1
Time: 17.3s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.1231080354848577279591737010377983096987:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{\frac{2}{1 + e^{-2 \cdot x}} - 1}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}}\\ \mathbf{elif}\;-2 \cdot x \le 2.985881592287361997487503412755010037682 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \sqrt[3]{{\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)}^{3}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.1231080354848577279591737010377983096987:\\
\;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{\frac{2}{1 + e^{-2 \cdot x}} - 1}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}}\\

\mathbf{elif}\;-2 \cdot x \le 2.985881592287361997487503412755010037682 \cdot 10^{-10}:\\
\;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}}\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r57135 = 2.0;
        double r57136 = 1.0;
        double r57137 = -2.0;
        double r57138 = x;
        double r57139 = r57137 * r57138;
        double r57140 = exp(r57139);
        double r57141 = r57136 + r57140;
        double r57142 = r57135 / r57141;
        double r57143 = r57142 - r57136;
        return r57143;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r57144 = -2.0;
        double r57145 = x;
        double r57146 = r57144 * r57145;
        double r57147 = -0.12310803548485773;
        bool r57148 = r57146 <= r57147;
        double r57149 = 2.0;
        double r57150 = 1.0;
        double r57151 = exp(r57146);
        double r57152 = r57150 + r57151;
        double r57153 = r57149 / r57152;
        double r57154 = sqrt(r57153);
        double r57155 = sqrt(r57150);
        double r57156 = r57154 + r57155;
        double r57157 = r57153 - r57150;
        double r57158 = r57157 / r57156;
        double r57159 = r57156 * r57158;
        double r57160 = 2.985881592287362e-10;
        bool r57161 = r57146 <= r57160;
        double r57162 = r57150 * r57145;
        double r57163 = 5.551115123125783e-17;
        double r57164 = 4.0;
        double r57165 = pow(r57145, r57164);
        double r57166 = r57163 * r57165;
        double r57167 = 0.33333333333333337;
        double r57168 = 3.0;
        double r57169 = pow(r57145, r57168);
        double r57170 = r57167 * r57169;
        double r57171 = r57166 + r57170;
        double r57172 = r57162 - r57171;
        double r57173 = r57172 / r57156;
        double r57174 = r57156 * r57173;
        double r57175 = r57154 - r57155;
        double r57176 = pow(r57175, r57168);
        double r57177 = cbrt(r57176);
        double r57178 = r57156 * r57177;
        double r57179 = r57161 ? r57174 : r57178;
        double r57180 = r57148 ? r57159 : r57179;
        return r57180;
}

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 3 regimes

  2. if (* -2.0 x) < -0.12310803548485773

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt1.6

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

    5. Applied difference-of-squares1.0

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

    7. Applied add-cbrt-cube1.0

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

    8. Simplified1.0

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

    10. Applied flip--1.6

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

    11. Applied cube-div1.6

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

    12. Applied cbrt-div2.3

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

    13. Simplified1.0

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

    14. Simplified0.0

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

    if -0.12310803548485773 < (* -2.0 x) < 2.985881592287362e-10

    1. Initial program 59.4

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

    3. Applied add-sqr-sqrt59.4

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

    4. Applied add-sqr-sqrt59.5

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

    5. Applied difference-of-squares59.5

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

    7. Applied add-cbrt-cube59.5

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

    8. Simplified59.5

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

    10. Applied flip--59.5

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

    11. Applied cube-div59.5

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

    12. Applied cbrt-div59.5

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

    13. Simplified59.4

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

    14. Simplified59.4

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

    15. Taylor expanded around 0 0.1

      \[\leadsto \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{\color{blue}{1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)}}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}}\]

    if 2.985881592287362e-10 < (* -2.0 x)

    1. Initial program 0.4

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

    3. Applied add-sqr-sqrt0.4

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

    4. Applied add-sqr-sqrt0.4

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

    5. Applied difference-of-squares0.4

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

    7. Applied add-cbrt-cube0.4

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

    8. Simplified0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.1231080354848577279591737010377983096987:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{\frac{2}{1 + e^{-2 \cdot x}} - 1}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}}\\ \mathbf{elif}\;-2 \cdot x \le 2.985881592287361997487503412755010037682 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \frac{1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \sqrt[3]{{\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))