Average Error: 29.2 → 0.2
Time: 3.6s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.010810868465119151:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)} \cdot \sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 4.707367496713552 \cdot 10^{-11}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.010810868465119151:\\
\;\;\;\;{\left(e^{\sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)} \cdot \sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)}\\

\mathbf{elif}\;-2 \cdot x \leq 4.707367496713552 \cdot 10^{-11}:\\
\;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\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) -0.010810868465119151)
   (pow
    (exp
     (*
      (cbrt (log (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
      (cbrt (log (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))))
    (cbrt (log (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))))
   (if (<= (* -2.0 x) 4.707367496713552e-11)
     (-
      (+ 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) <= -0.010810868465119151) {
		tmp = pow(exp(cbrt(log((2.0 / (1.0 + exp(-2.0 * x))) - 1.0)) * cbrt(log((2.0 / (1.0 + exp(-2.0 * x))) - 1.0))), cbrt(log((2.0 / (1.0 + exp(-2.0 * x))) - 1.0)));
	} else if ((-2.0 * x) <= 4.707367496713552e-11) {
		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) < -0.010810868465119151

    1. Initial program 0.0

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

    3. Applied add-exp-log_binary64_4490.0

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

    5. Applied add-cube-cbrt_binary64_4460.0

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

    6. Applied exp-prod_binary64_4630.0

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

    if -0.010810868465119151 < (*.f64 -2 x) < 4.70736749671355203e-11

    1. Initial program 59.5

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

    2. Taylor expanded around 0 0.0

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

    if 4.70736749671355203e-11 < (*.f64 -2 x)

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt_binary64_4350.6

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

    4. Applied associate-/r*_binary64_3600.6

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.010810868465119151:\\ \;\;\;\;{\left(e^{\sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)} \cdot \sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}}\right)}^{\left(\sqrt[3]{\log \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 4.707367496713552 \cdot 10^{-11}:\\ \;\;\;\;\left(x + 0.13333333333333333 \cdot {x}^{5}\right) - 0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))