Logistic function from Lakshay Garg

Average Error: 29.0 → 0.0

Time: 51.4s

Precision: 64

• unused variable y

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.007903465556076343:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.00792720968856004:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{-1}{3} \cdot x\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{2}{15}, \left({x}^{5}\right), x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

C

double f(double x, double __attribute__((unused)) y) {
        double r10055667 = 2.0;
        double r10055668 = 1.0;
        double r10055669 = -2.0;
        double r10055670 = x;
        double r10055671 = r10055669 * r10055670;
        double r10055672 = exp(r10055671);
        double r10055673 = r10055668 + r10055672;
        double r10055674 = r10055667 / r10055673;
        double r10055675 = r10055674 - r10055668;
        return r10055675;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r10055676 = x;
        double r10055677 = -0.007903465556076343;
        bool r10055678 = r10055676 <= r10055677;
        double r10055679 = 2.0;
        double r10055680 = -2.0;
        double r10055681 = r10055680 * r10055676;
        double r10055682 = exp(r10055681);
        double r10055683 = 1.0;
        double r10055684 = r10055682 + r10055683;
        double r10055685 = r10055679 / r10055684;
        double r10055686 = r10055685 - r10055683;
        double r10055687 = 0.00792720968856004;
        bool r10055688 = r10055676 <= r10055687;
        double r10055689 = -0.3333333333333333;
        double r10055690 = r10055689 * r10055676;
        double r10055691 = r10055676 * r10055676;
        double r10055692 = 0.13333333333333333;
        double r10055693 = 5.0;
        double r10055694 = pow(r10055676, r10055693);
        double r10055695 = fma(r10055692, r10055694, r10055676);
        double r10055696 = fma(r10055690, r10055691, r10055695);
        double r10055697 = r10055688 ? r10055696 : r10055686;
        double r10055698 = r10055678 ? r10055686 : r10055697;
        return r10055698;
}

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if x < -0.007903465556076343 or 0.00792720968856004 < x

   1. Initial program 0.0

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   if -0.007903465556076343 < x < 0.00792720968856004

   1. Initial program 59.0

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

   2. Taylor expanded around inf 59.0

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

   3. Simplified 59.0

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

   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

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007903465556076343:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.00792720968856004:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{-1}{3} \cdot x\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\frac{2}{15}, \left({x}^{5}\right), x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))