Average Error: 29.2 → 0.0
Time: 11.5s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.0132325776020981046:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\ \mathbf{elif}\;-2 \cdot x \le 2.72138255369031824 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.0132325776020981046:\\
\;\;\;\;\left(\sqrt[3]{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\

\mathbf{elif}\;-2 \cdot x \le 2.72138255369031824 \cdot 10^{-5}:\\
\;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r59401 = 2.0;
        double r59402 = 1.0;
        double r59403 = -2.0;
        double r59404 = x;
        double r59405 = r59403 * r59404;
        double r59406 = exp(r59405);
        double r59407 = r59402 + r59406;
        double r59408 = r59401 / r59407;
        double r59409 = r59408 - r59402;
        return r59409;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r59410 = -2.0;
        double r59411 = x;
        double r59412 = r59410 * r59411;
        double r59413 = -0.013232577602098105;
        bool r59414 = r59412 <= r59413;
        double r59415 = 2.0;
        double r59416 = 1.0;
        double r59417 = exp(r59412);
        double r59418 = r59416 + r59417;
        double r59419 = r59415 / r59418;
        double r59420 = r59419 * r59419;
        double r59421 = r59416 * r59416;
        double r59422 = r59420 - r59421;
        double r59423 = r59419 + r59416;
        double r59424 = r59422 / r59423;
        double r59425 = cbrt(r59424);
        double r59426 = r59419 - r59416;
        double r59427 = cbrt(r59426);
        double r59428 = r59425 * r59427;
        double r59429 = r59428 * r59427;
        double r59430 = 2.7213825536903182e-05;
        bool r59431 = r59412 <= r59430;
        double r59432 = r59416 * r59411;
        double r59433 = 5.551115123125783e-17;
        double r59434 = 4.0;
        double r59435 = pow(r59411, r59434);
        double r59436 = r59433 * r59435;
        double r59437 = 0.33333333333333337;
        double r59438 = 3.0;
        double r59439 = pow(r59411, r59438);
        double r59440 = r59437 * r59439;
        double r59441 = r59436 + r59440;
        double r59442 = r59432 - r59441;
        double r59443 = cbrt(r59418);
        double r59444 = r59443 * r59443;
        double r59445 = r59444 * r59443;
        double r59446 = r59415 / r59445;
        double r59447 = r59446 - r59416;
        double r59448 = r59431 ? r59442 : r59447;
        double r59449 = r59414 ? r59429 : r59448;
        return r59449;
}

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) < -0.013232577602098105

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    5. Applied flip--0.0

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

    if -0.013232577602098105 < (* -2.0 x) < 2.7213825536903182e-05

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)}\]

    if 2.7213825536903182e-05 < (* -2.0 x)

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.0132325776020981046:\\ \;\;\;\;\left(\sqrt[3]{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} - 1}\\ \mathbf{elif}\;-2 \cdot x \le 2.72138255369031824 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot x - \left(5.55112 \cdot 10^{-17} \cdot {x}^{4} + 0.33333333333333337 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{1 + e^{-2 \cdot x}} \cdot \sqrt[3]{1 + e^{-2 \cdot x}}\right) \cdot \sqrt[3]{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))