Logistic function from Lakshay Garg

Average Error: 29.1 → 0.1

Time: 33.3s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.008574052355382087:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;-2 \cdot x \le 1.4278659301426067 \cdot 10^{-07}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r2212689 = 2.0;
        double r2212690 = 1.0;
        double r2212691 = -2.0;
        double r2212692 = x;
        double r2212693 = r2212691 * r2212692;
        double r2212694 = exp(r2212693);
        double r2212695 = r2212690 + r2212694;
        double r2212696 = r2212689 / r2212695;
        double r2212697 = r2212696 - r2212690;
        return r2212697;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r2212698 = -2.0;
        double r2212699 = x;
        double r2212700 = r2212698 * r2212699;
        double r2212701 = -0.008574052355382087;
        bool r2212702 = r2212700 <= r2212701;
        double r2212703 = 2.0;
        double r2212704 = exp(r2212700);
        double r2212705 = 1.0;
        double r2212706 = r2212704 + r2212705;
        double r2212707 = r2212703 / r2212706;
        double r2212708 = r2212707 - r2212705;
        double r2212709 = 1.4278659301426067e-07;
        bool r2212710 = r2212700 <= r2212709;
        double r2212711 = r2212699 * r2212699;
        double r2212712 = r2212699 * r2212711;
        double r2212713 = -0.3333333333333333;
        double r2212714 = 5.0;
        double r2212715 = pow(r2212699, r2212714);
        double r2212716 = 0.13333333333333333;
        double r2212717 = fma(r2212715, r2212716, r2212699);
        double r2212718 = fma(r2212712, r2212713, r2212717);
        double r2212719 = r2212710 ? r2212718 : r2212708;
        double r2212720 = r2212702 ? r2212708 : r2212719;
        return r2212720;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (* -2 x) < -0.008574052355382087 or 1.4278659301426067e-07 < (* -2 x)

   1. Initial program 0.1

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

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

   if -0.008574052355382087 < (* -2 x) < 1.4278659301426067e-07

   1. Initial program 59.3

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

   2. Taylor expanded around inf 59.3

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

   3. Simplified 59.3

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

   4. Taylor expanded around 0 0.0

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

   5. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.008574052355382087:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;-2 \cdot x \le 1.4278659301426067 \cdot 10^{-07}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-1}{3}, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))