Average Error: 30.0 → 0.0
Time: 2.2m
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007294245055471922:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.008176101527675436:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.007294245055471922:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{elif}\;x \le 0.008176101527675436:\\
\;\;\;\;(\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}{1 + e^{-2 \cdot x}} - 1\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r7250394 = 2.0;
        double r7250395 = 1.0;
        double r7250396 = -2.0;
        double r7250397 = x;
        double r7250398 = r7250396 * r7250397;
        double r7250399 = exp(r7250398);
        double r7250400 = r7250395 + r7250399;
        double r7250401 = r7250394 / r7250400;
        double r7250402 = r7250401 - r7250395;
        return r7250402;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r7250403 = x;
        double r7250404 = -0.007294245055471922;
        bool r7250405 = r7250403 <= r7250404;
        double r7250406 = 2.0;
        double r7250407 = 1.0;
        double r7250408 = -2.0;
        double r7250409 = r7250408 * r7250403;
        double r7250410 = exp(r7250409);
        double r7250411 = r7250407 + r7250410;
        double r7250412 = r7250406 / r7250411;
        double r7250413 = r7250412 - r7250407;
        double r7250414 = 0.008176101527675436;
        bool r7250415 = r7250403 <= r7250414;
        double r7250416 = -0.3333333333333333;
        double r7250417 = r7250416 * r7250403;
        double r7250418 = r7250403 * r7250403;
        double r7250419 = 0.13333333333333333;
        double r7250420 = 5.0;
        double r7250421 = pow(r7250403, r7250420);
        double r7250422 = fma(r7250419, r7250421, r7250403);
        double r7250423 = fma(r7250417, r7250418, r7250422);
        double r7250424 = r7250415 ? r7250423 : r7250413;
        double r7250425 = r7250405 ? r7250413 : r7250424;
        return r7250425;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.007294245055471922 or 0.008176101527675436 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    if -0.007294245055471922 < x < 0.008176101527675436

    1. Initial program 59.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007294245055471922:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.008176101527675436:\\ \;\;\;\;(\left(\frac{-1}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\frac{2}{15} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))