Average Error: 29.0 → 0.1
Time: 23.5s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\
\;\;\;\;\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\

\mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\
\;\;\;\;\left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r2760422 = 2.0;
        double r2760423 = 1.0;
        double r2760424 = -2.0;
        double r2760425 = x;
        double r2760426 = r2760424 * r2760425;
        double r2760427 = exp(r2760426);
        double r2760428 = r2760423 + r2760427;
        double r2760429 = r2760422 / r2760428;
        double r2760430 = r2760429 - r2760423;
        return r2760430;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r2760431 = -2.0;
        double r2760432 = x;
        double r2760433 = r2760431 * r2760432;
        double r2760434 = -15.661220476650124;
        bool r2760435 = r2760433 <= r2760434;
        double r2760436 = 2.0;
        double r2760437 = 1.0;
        double r2760438 = exp(r2760433);
        double r2760439 = r2760437 + r2760438;
        double r2760440 = r2760436 / r2760439;
        double r2760441 = r2760440 - r2760437;
        double r2760442 = r2760441 * r2760441;
        double r2760443 = r2760441 * r2760442;
        double r2760444 = cbrt(r2760443);
        double r2760445 = 5.846791957771714e-06;
        bool r2760446 = r2760433 <= r2760445;
        double r2760447 = 0.33333333333333337;
        double r2760448 = r2760432 * r2760432;
        double r2760449 = r2760447 * r2760448;
        double r2760450 = r2760437 - r2760449;
        double r2760451 = r2760450 * r2760432;
        double r2760452 = 5.551115123125783e-17;
        double r2760453 = r2760448 * r2760452;
        double r2760454 = r2760448 * r2760453;
        double r2760455 = r2760451 - r2760454;
        double r2760456 = r2760446 ? r2760455 : r2760444;
        double r2760457 = r2760435 ? r2760444 : r2760456;
        return r2760457;
}

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 (* -2.0 x) < -15.661220476650124 or 5.846791957771714e-06 < (* -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 associate-/r*0.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}}} - 1\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.1

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

    7. Simplified0.1

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

    if -15.661220476650124 < (* -2.0 x) < 5.846791957771714e-06

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

      \[\leadsto \color{blue}{x \cdot \left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - 0.3333333333333333703407674875052180141211 \cdot \left(x \cdot x\right)\right) \cdot x - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 5.5511151231257827021181583404541015625 \cdot 10^{-17}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))