Average Error: 29.2 → 0.1
Time: 27.3s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.3818320007707001750851816268550464883447:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)}\right)}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.537410466881136385719139325622961678164 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.3818320007707001750851816268550464883447:\\
\;\;\;\;\left(\sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\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)}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\

\mathbf{elif}\;-2 \cdot x \le 1.537410466881136385719139325622961678164 \cdot 10^{-5}:\\
\;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r4488473 = 2.0;
        double r4488474 = 1.0;
        double r4488475 = -2.0;
        double r4488476 = x;
        double r4488477 = r4488475 * r4488476;
        double r4488478 = exp(r4488477);
        double r4488479 = r4488474 + r4488478;
        double r4488480 = r4488473 / r4488479;
        double r4488481 = r4488480 - r4488474;
        return r4488481;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r4488482 = -2.0;
        double r4488483 = x;
        double r4488484 = r4488482 * r4488483;
        double r4488485 = -0.3818320007707002;
        bool r4488486 = r4488484 <= r4488485;
        double r4488487 = 2.0;
        double r4488488 = 1.0;
        double r4488489 = exp(r4488484);
        double r4488490 = r4488488 + r4488489;
        double r4488491 = r4488487 / r4488490;
        double r4488492 = r4488491 - r4488488;
        double r4488493 = exp(r4488492);
        double r4488494 = log(r4488493);
        double r4488495 = cbrt(r4488494);
        double r4488496 = sqrt(r4488491);
        double r4488497 = sqrt(r4488488);
        double r4488498 = r4488496 + r4488497;
        double r4488499 = r4488496 - r4488497;
        double r4488500 = r4488498 * r4488499;
        double r4488501 = exp(r4488500);
        double r4488502 = log(r4488501);
        double r4488503 = cbrt(r4488502);
        double r4488504 = r4488495 * r4488503;
        double r4488505 = cbrt(r4488492);
        double r4488506 = r4488504 * r4488505;
        double r4488507 = 1.5374104668811364e-05;
        bool r4488508 = r4488484 <= r4488507;
        double r4488509 = r4488483 * r4488483;
        double r4488510 = 0.33333333333333337;
        double r4488511 = r4488509 * r4488510;
        double r4488512 = r4488488 - r4488511;
        double r4488513 = r4488483 * r4488512;
        double r4488514 = 5.551115123125783e-17;
        double r4488515 = r4488509 * r4488509;
        double r4488516 = r4488514 * r4488515;
        double r4488517 = r4488513 - r4488516;
        double r4488518 = 1.0;
        double r4488519 = sqrt(r4488490);
        double r4488520 = r4488518 / r4488519;
        double r4488521 = r4488487 / r4488519;
        double r4488522 = r4488520 * r4488521;
        double r4488523 = r4488522 - r4488488;
        double r4488524 = r4488508 ? r4488517 : r4488523;
        double r4488525 = r4488486 ? r4488506 : r4488524;
        return r4488525;
}

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) < -0.3818320007707002

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    8. Applied add-cube-cbrt0.0

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

    10. Applied exp-diff0.0

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

    11. Applied log-div0.0

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

    12. Simplified0.0

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

    13. Simplified0.0

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

    15. Applied add-sqr-sqrt0.0

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

    16. Applied add-sqr-sqrt0.0

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

    17. Applied difference-of-squares0.0

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

    if -0.3818320007707002 < (* -2.0 x) < 1.5374104668811364e-05

    1. Initial program 59.3

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if 1.5374104668811364e-05 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied *-un-lft-identity0.1

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

    5. Applied times-frac0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.3818320007707001750851816268550464883447:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)}\right)}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.537410466881136385719139325622961678164 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + e^{-2 \cdot x}}} \cdot \frac{2}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))