Logistic function from Lakshay Garg

Average Error: 29.2 → 0.5

Time: 15.7s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -65281272.71744271:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 0.0006247476447836436:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{3}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r834019 = 2.0;
        double r834020 = 1.0;
        double r834021 = -2.0;
        double r834022 = x;
        double r834023 = r834021 * r834022;
        double r834024 = exp(r834023);
        double r834025 = r834020 + r834024;
        double r834026 = r834019 / r834025;
        double r834027 = r834026 - r834020;
        return r834027;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r834028 = -2.0;
        double r834029 = x;
        double r834030 = r834028 * r834029;
        double r834031 = -65281272.71744271;
        bool r834032 = r834030 <= r834031;
        double r834033 = 2.0;
        double r834034 = exp(r834030);
        double r834035 = 1.0;
        double r834036 = r834034 + r834035;
        double r834037 = sqrt(r834036);
        double r834038 = r834033 / r834037;
        double r834039 = r834038 / r834037;
        double r834040 = r834039 - r834035;
        double r834041 = 0.0006247476447836436;
        bool r834042 = r834030 <= r834041;
        double r834043 = -0.3333333333333333;
        double r834044 = r834029 * r834029;
        double r834045 = r834044 * r834029;
        double r834046 = 5.0;
        double r834047 = pow(r834029, r834046);
        double r834048 = 0.13333333333333333;
        double r834049 = fma(r834047, r834048, r834029);
        double r834050 = fma(r834043, r834045, r834049);
        double r834051 = r834042 ? r834050 : r834040;
        double r834052 = r834032 ? r834040 : r834051;
        return r834052;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (* -2 x) < -65281272.71744271 or 0.0006247476447836436 < (* -2 x)

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt 0.0

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

   4. Applied associate-/r* 0.0

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

   if -65281272.71744271 < (* -2 x) < 0.0006247476447836436

   1. Initial program 58.0

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

   2. Taylor expanded around 0 0.9

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

   3. Simplified 0.9

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -65281272.71744271:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 0.0006247476447836436:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{3}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))