Average Error: 29.2 → 0.2
Time: 3.8s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1599.2511333305727:\\ \;\;\;\;\frac{2}{\frac{1 + {\left(e^{x}\right)}^{-6}}{1 + \left({\left(e^{x}\right)}^{-4} - e^{-2 \cdot x}\right)}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 6.477663099114865 \cdot 10^{-06}:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{1 + e^{-2 \cdot x}} \cdot \sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -1599.2511333305727:\\
\;\;\;\;\frac{2}{\frac{1 + {\left(e^{x}\right)}^{-6}}{1 + \left({\left(e^{x}\right)}^{-4} - e^{-2 \cdot x}\right)}} - 1\\

\mathbf{elif}\;-2 \cdot x \leq 6.477663099114865 \cdot 10^{-06}:\\
\;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\sqrt{1 + e^{-2 \cdot x}} \cdot \sqrt{1 + e^{-2 \cdot x}}} - 1\\

\end{array}
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -1599.2511333305727)
   (-
    (/
     2.0
     (/
      (+ 1.0 (pow (exp x) -6.0))
      (+ 1.0 (- (pow (exp x) -4.0) (exp (* -2.0 x))))))
    1.0)
   (if (<= (* -2.0 x) 6.477663099114865e-06)
     (+
      x
      (-
       (* 0.13333333333333333 (pow x 5.0))
       (* 0.3333333333333333 (pow x 3.0))))
     (-
      (/
       2.0
       (* (sqrt (+ 1.0 (exp (* -2.0 x)))) (sqrt (+ 1.0 (exp (* -2.0 x))))))
      1.0))))
double code(double x, double y) {
	return (2.0 / (1.0 + exp(-2.0 * x))) - 1.0;
}
↓
double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -1599.2511333305727) {
		tmp = (2.0 / ((1.0 + pow(exp(x), -6.0)) / (1.0 + (pow(exp(x), -4.0) - exp(-2.0 * x))))) - 1.0;
	} else if ((-2.0 * x) <= 6.477663099114865e-06) {
		tmp = x + ((0.13333333333333333 * pow(x, 5.0)) - (0.3333333333333333 * pow(x, 3.0)));
	} else {
		tmp = (2.0 / (sqrt(1.0 + exp(-2.0 * x)) * sqrt(1.0 + exp(-2.0 * x)))) - 1.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 -2 x) < -1599.25113333057266

    1. Initial program 0

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

    3. Applied flip3-+_binary64_31500

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

    4. Simplified0

      \[\leadsto \frac{2}{\frac{\color{blue}{1 + {\left(e^{x}\right)}^{-6}}}{1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)}} - 1\]

    5. Simplified0

      \[\leadsto \frac{2}{\frac{1 + {\left(e^{x}\right)}^{-6}}{\color{blue}{1 + \left({\left(e^{x}\right)}^{-4} - e^{-2 \cdot x}\right)}}} - 1\]

    if -1599.25113333057266 < (*.f64 -2 x) < 6.4776630991148652e-6

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}}\]
    3. Using strategy rm

    4. Applied associate--l+_binary64_30840.3

      \[\leadsto \color{blue}{x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)}\]

    if 6.4776630991148652e-6 < (*.f64 -2 x)

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt_binary64_31690.1

      \[\leadsto \frac{2}{\color{blue}{\sqrt{1 + e^{-2 \cdot x}} \cdot \sqrt{1 + e^{-2 \cdot x}}}} - 1\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1599.2511333305727:\\ \;\;\;\;\frac{2}{\frac{1 + {\left(e^{x}\right)}^{-6}}{1 + \left({\left(e^{x}\right)}^{-4} - e^{-2 \cdot x}\right)}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 6.477663099114865 \cdot 10^{-06}:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} - 0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{1 + e^{-2 \cdot x}} \cdot \sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))