Logistic function from Lakshay Garg

Average Error: 29.8 → 0.0

Time: 13.8s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.0485854444297498:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 0.0031955453096448086:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{3}, x \cdot \left(x \cdot x\right), \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r1250465 = 2.0;
        double r1250466 = 1.0;
        double r1250467 = -2.0;
        double r1250468 = x;
        double r1250469 = r1250467 * r1250468;
        double r1250470 = exp(r1250469);
        double r1250471 = r1250466 + r1250470;
        double r1250472 = r1250465 / r1250471;
        double r1250473 = r1250472 - r1250466;
        return r1250473;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r1250474 = -2.0;
        double r1250475 = x;
        double r1250476 = r1250474 * r1250475;
        double r1250477 = -1.0485854444297498;
        bool r1250478 = r1250476 <= r1250477;
        double r1250479 = 2.0;
        double r1250480 = 1.0;
        double r1250481 = exp(r1250476);
        double r1250482 = r1250480 + r1250481;
        double r1250483 = r1250479 / r1250482;
        double r1250484 = r1250483 - r1250480;
        double r1250485 = 0.0031955453096448086;
        bool r1250486 = r1250476 <= r1250485;
        double r1250487 = -0.3333333333333333;
        double r1250488 = r1250475 * r1250475;
        double r1250489 = r1250475 * r1250488;
        double r1250490 = 5.0;
        double r1250491 = pow(r1250475, r1250490);
        double r1250492 = 0.13333333333333333;
        double r1250493 = fma(r1250491, r1250492, r1250475);
        double r1250494 = fma(r1250487, r1250489, r1250493);
        double r1250495 = r1250486 ? r1250494 : r1250484;
        double r1250496 = r1250478 ? r1250484 : r1250495;
        return r1250496;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (* -2 x) < -1.0485854444297498 or 0.0031955453096448086 < (* -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

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

   5. Taylor expanded around inf 0.0

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

   if -1.0485854444297498 < (* -2 x) < 0.0031955453096448086

   1. Initial program 59.0

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

   2. Taylor expanded around 0 0.1

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

   3. Simplified 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1.0485854444297498:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 0.0031955453096448086:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{3}, x \cdot \left(x \cdot x\right), \mathsf{fma}\left({x}^{5}, \frac{2}{15}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))