Average Error: 29.4 → 0.0
Time: 54.1s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007333471846547681:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.006247673021062525:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.007333471846547681:\\
\;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\

\mathbf{elif}\;x \le 0.006247673021062525:\\
\;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r4779573 = 2.0;
        double r4779574 = 1.0;
        double r4779575 = -2.0;
        double r4779576 = x;
        double r4779577 = r4779575 * r4779576;
        double r4779578 = exp(r4779577);
        double r4779579 = r4779574 + r4779578;
        double r4779580 = r4779573 / r4779579;
        double r4779581 = r4779580 - r4779574;
        return r4779581;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r4779582 = x;
        double r4779583 = -0.007333471846547681;
        bool r4779584 = r4779582 <= r4779583;
        double r4779585 = 2.0;
        double r4779586 = -2.0;
        double r4779587 = r4779586 * r4779582;
        double r4779588 = exp(r4779587);
        double r4779589 = 1.0;
        double r4779590 = r4779588 + r4779589;
        double r4779591 = r4779585 / r4779590;
        double r4779592 = r4779591 - r4779589;
        double r4779593 = 0.006247673021062525;
        bool r4779594 = r4779582 <= r4779593;
        double r4779595 = -0.3333333333333333;
        double r4779596 = r4779595 * r4779582;
        double r4779597 = r4779582 * r4779582;
        double r4779598 = 0.13333333333333333;
        double r4779599 = 5.0;
        double r4779600 = pow(r4779582, r4779599);
        double r4779601 = fma(r4779598, r4779600, r4779582);
        double r4779602 = fma(r4779596, r4779597, r4779601);
        double r4779603 = r4779594 ? r4779602 : r4779592;
        double r4779604 = r4779584 ? r4779592 : r4779603;
        return r4779604;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.007333471846547681 or 0.006247673021062525 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -0.007333471846547681 < x < 0.006247673021062525

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{(\left(x \cdot \frac{-1}{3}\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007333471846547681:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.006247673021062525:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))