Logistic function from Lakshay Garg

Average Error: 29.6 → 0.2

Time: 39.7s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -83.20148025804453:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;-2 \cdot x \le 3.58822878881946 \cdot 10^{-05}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{15}, {x}^{5}, \mathsf{fma}\left(\frac{-1}{3} \cdot x, x \cdot x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r1977626 = 2.0;
        double r1977627 = 1.0;
        double r1977628 = -2.0;
        double r1977629 = x;
        double r1977630 = r1977628 * r1977629;
        double r1977631 = exp(r1977630);
        double r1977632 = r1977627 + r1977631;
        double r1977633 = r1977626 / r1977632;
        double r1977634 = r1977633 - r1977627;
        return r1977634;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r1977635 = -2.0;
        double r1977636 = x;
        double r1977637 = r1977635 * r1977636;
        double r1977638 = -83.20148025804453;
        bool r1977639 = r1977637 <= r1977638;
        double r1977640 = 2.0;
        double r1977641 = exp(r1977637);
        double r1977642 = 1.0;
        double r1977643 = r1977641 + r1977642;
        double r1977644 = r1977640 / r1977643;
        double r1977645 = r1977644 - r1977642;
        double r1977646 = 3.58822878881946e-05;
        bool r1977647 = r1977637 <= r1977646;
        double r1977648 = 0.13333333333333333;
        double r1977649 = 5.0;
        double r1977650 = pow(r1977636, r1977649);
        double r1977651 = -0.3333333333333333;
        double r1977652 = r1977651 * r1977636;
        double r1977653 = r1977636 * r1977636;
        double r1977654 = fma(r1977652, r1977653, r1977636);
        double r1977655 = fma(r1977648, r1977650, r1977654);
        double r1977656 = r1977647 ? r1977655 : r1977645;
        double r1977657 = r1977639 ? r1977645 : r1977656;
        return r1977657;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (* -2 x) < -83.20148025804453 or 3.58822878881946e-05 < (* -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 -83.20148025804453 < (* -2 x) < 3.58822878881946e-05

   1. Initial program 59.1

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

   2. Taylor expanded around 0 0.2

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

   3. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -83.20148025804453:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;-2 \cdot x \le 3.58822878881946 \cdot 10^{-05}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{15}, {x}^{5}, \mathsf{fma}\left(\frac{-1}{3} \cdot x, x \cdot x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))