Logistic function from Lakshay Garg

Average Error: 29.1 → 0.1

Time: 44.8s

Precision: 64

• unused variable y

Math

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

↓

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

C

double f(double x, double __attribute__((unused)) y) {
        double r2016023 = 2.0;
        double r2016024 = 1.0;
        double r2016025 = -2.0;
        double r2016026 = x;
        double r2016027 = r2016025 * r2016026;
        double r2016028 = exp(r2016027);
        double r2016029 = r2016024 + r2016028;
        double r2016030 = r2016023 / r2016029;
        double r2016031 = r2016030 - r2016024;
        return r2016031;
}

↓

double f(double x, double __attribute__((unused)) y) {
        double r2016032 = -2.0;
        double r2016033 = x;
        double r2016034 = r2016032 * r2016033;
        double r2016035 = -0.008574052355382087;
        bool r2016036 = r2016034 <= r2016035;
        double r2016037 = 2.0;
        double r2016038 = exp(r2016034);
        double r2016039 = 1.0;
        double r2016040 = r2016038 + r2016039;
        double r2016041 = r2016037 / r2016040;
        double r2016042 = r2016041 - r2016039;
        double r2016043 = 1.4278659301426067e-07;
        bool r2016044 = r2016034 <= r2016043;
        double r2016045 = 0.13333333333333333;
        double r2016046 = 5.0;
        double r2016047 = pow(r2016033, r2016046);
        double r2016048 = r2016045 * r2016047;
        double r2016049 = 0.3333333333333333;
        double r2016050 = r2016049 * r2016033;
        double r2016051 = r2016033 * r2016050;
        double r2016052 = r2016051 * r2016033;
        double r2016053 = r2016048 - r2016052;
        double r2016054 = r2016053 + r2016033;
        double r2016055 = r2016044 ? r2016054 : r2016042;
        double r2016056 = r2016036 ? r2016042 : r2016055;
        return r2016056;
}

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

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

4. Final simplification 0.1

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

Reproduce

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