Average Error: 29.1 → 0.1
Time: 4.2s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1.5465792801287923:\\ \;\;\;\;\frac{\frac{-1 + \frac{16}{{\left(1 + e^{-2 \cdot x}\right)}^{4}}}{1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 1.065631806124204 \cdot 10^{-08}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -1.5465792801287923:\\
\;\;\;\;\frac{\frac{-1 + \frac{16}{{\left(1 + e^{-2 \cdot x}\right)}^{4}}}{1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\

\mathbf{elif}\;-2 \cdot x \leq 1.065631806124204 \cdot 10^{-08}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 + \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}}}\\

\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) -1.5465792801287923)
   (/
    (/
     (+ -1.0 (/ 16.0 (pow (+ 1.0 (exp (* -2.0 x))) 4.0)))
     (+ 1.0 (/ 4.0 (pow (+ 1.0 (exp (* -2.0 x))) 2.0))))
    (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) 1.065631806124204e-08)
     (-
      (+ x (* 0.13333333333333333 (pow x 5.0)))
      (* 0.3333333333333333 (pow x 3.0)))
     (/
      (+ -1.0 (/ 4.0 (* (+ 1.0 (exp (* -2.0 x))) (+ 1.0 (exp (* -2.0 x))))))
      (+ 1.0 (/ 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) <= -1.5465792801287923) {
		tmp = ((-1.0 + (16.0 / pow((1.0 + exp(-2.0 * x)), 4.0))) / (1.0 + (4.0 / pow((1.0 + exp(-2.0 * x)), 2.0)))) / (1.0 + (2.0 / (1.0 + exp(-2.0 * x))));
	} else if ((-2.0 * x) <= 1.065631806124204e-08) {
		tmp = (x + (0.13333333333333333 * pow(x, 5.0))) - (0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = (-1.0 + (4.0 / ((1.0 + exp(-2.0 * x)) * (1.0 + exp(-2.0 * x))))) / (1.0 + (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) < -1.54657928012879231

    1. Initial program 0.0

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

    3. Applied flip--_binary64_3980.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    7. Applied flip-+_binary64_3970.0

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

    8. Simplified0.0

      \[\leadsto \frac{\frac{\color{blue}{-1 + \frac{16}{{\left(1 + e^{-2 \cdot x}\right)}^{4}}}}{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} - -1}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\]

    9. Simplified0.0

      \[\leadsto \frac{\frac{-1 + \frac{16}{{\left(1 + e^{-2 \cdot x}\right)}^{4}}}{\color{blue}{1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\]

    if -1.54657928012879231 < (*.f64 -2 x) < 1.06563180612420405e-8

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.1

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

    if 1.06563180612420405e-8 < (*.f64 -2 x)

    1. Initial program 0.3

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

    3. Applied flip--_binary64_3980.3

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

    4. Simplified0.3

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

    5. Simplified0.3

      \[\leadsto \frac{\frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + -1}{\color{blue}{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1.5465792801287923:\\ \;\;\;\;\frac{\frac{-1 + \frac{16}{{\left(1 + e^{-2 \cdot x}\right)}^{4}}}{1 + \frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}}}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 1.065631806124204 \cdot 10^{-08}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{4}{\left(1 + e^{-2 \cdot x}\right) \cdot \left(1 + e^{-2 \cdot x}\right)}}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))