Average Error: 29.6 → 0.8
Time: 3.1s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5.402290062798888 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 4.5280989964747235 \cdot 10^{-07}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(-1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -5.402290062798888 \cdot 10^{+16}:\\
\;\;\;\;\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\

\mathbf{elif}\;-2 \cdot x \leq 4.5280989964747235 \cdot 10^{-07}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

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

\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) -5.402290062798888e+16)
   (/
    (+ (/ 4.0 (* (+ 1.0 (exp (* -2.0 x))) (+ 1.0 (exp (* -2.0 x))))) -1.0)
    (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) 4.5280989964747235e-07)
     (-
      (+ x (* 0.13333333333333333 (pow x 5.0)))
      (* 0.3333333333333333 (pow x 3.0)))
     (*
      (+ 1.0 (sqrt (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
      (+ -1.0 (sqrt (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))))))
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) <= -5.402290062798888e+16) {
		tmp = ((4.0 / ((1.0 + exp(-2.0 * x)) * (1.0 + exp(-2.0 * x)))) + -1.0) / (1.0 + (2.0 / (1.0 + exp(-2.0 * x))));
	} else if ((-2.0 * x) <= 4.5280989964747235e-07) {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = (1.0 + sqrt(2.0 / (1.0 + exp(-2.0 * x)))) * (-1.0 + sqrt(2.0 / (1.0 + exp(-2.0 * x))));
	}
	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) < -54022900627988880

    1. Initial program 0

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

    3. Applied flip--_binary64_530

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

    4. Simplified0

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

    5. Simplified0

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

    if -54022900627988880 < (*.f64 -2 x) < 4.5280989964747235e-7

    1. Initial program 57.8

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

    2. Taylor expanded around 0 1.5

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

    if 4.5280989964747235e-7 < (*.f64 -2 x)

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt_binary64_1000.2

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

    4. Applied difference-of-sqr-1_binary64_480.2

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

    5. Simplified0.2

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

    6. Simplified0.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5.402290062798888 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 4.5280989964747235 \cdot 10^{-07}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right) \cdot \left(-1 + \sqrt{\frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))