Logistic function from Lakshay Garg

Average Accuracy: 54.2% → 99.7%

Time: 14.3s

Precision: binary64

Cost: 53572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{\frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{2}{1 + e^{\left(-x\right) - x}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.1)
   (/
    (/
     (+ (* 4.0 (/ 1.0 (pow (+ 1.0 (exp (* -2.0 x))) 2.0))) -1.0)
     (pow (cbrt (+ 1.0 (/ 2.0 (+ 1.0 (exp (- (- x) x)))))) 2.0))
    (cbrt (+ 1.0 (/ 2.0 (+ 1.0 (pow (exp -2.0) x))))))
   (if (<= (* -2.0 x) 0.002)
     (+
      (* -0.3333333333333333 (pow x 3.0))
      (+ x (* 0.13333333333333333 (pow x 5.0))))
     -1.0)))

C

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) <= -0.1) {
		tmp = (((4.0 * (1.0 / pow((1.0 + exp((-2.0 * x))), 2.0))) + -1.0) / pow(cbrt((1.0 + (2.0 / (1.0 + exp((-x - x)))))), 2.0)) / cbrt((1.0 + (2.0 / (1.0 + pow(exp(-2.0), x)))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

↓

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = (((4.0 * (1.0 / Math.pow((1.0 + Math.exp((-2.0 * x))), 2.0))) + -1.0) / Math.pow(Math.cbrt((1.0 + (2.0 / (1.0 + Math.exp((-x - x)))))), 2.0)) / Math.cbrt((1.0 + (2.0 / (1.0 + Math.pow(Math.exp(-2.0), x)))));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

↓

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = Float64(Float64(Float64(Float64(4.0 * Float64(1.0 / (Float64(1.0 + exp(Float64(-2.0 * x))) ^ 2.0))) + -1.0) / (cbrt(Float64(1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(Float64(-x) - x)))))) ^ 2.0)) / cbrt(Float64(1.0 + Float64(2.0 / Float64(1.0 + (exp(-2.0) ^ x))))));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[(N[(N[(N[(4.0 * N[(1.0 / N[Power[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Power[N[Power[N[(1.0 + N[(2.0 / N[(1.0 + N[Exp[N[((-x) - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 + N[(2.0 / N[(1.0 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.10000000000000001

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \cdot \sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}} \]
      Proof

      [Start] 100.0	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      flip-- [=>] 100.0	\[ \color{blue}{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\frac{2}{1 + e^{-2 \cdot x}} + 1}} \]
      add-cube-cbrt [=>] 100.0	\[ \frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\color{blue}{\left(\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}\right) \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}}} \]
      associate-/r* [=>] 100.0	\[ \color{blue}{\frac{\frac{\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}} - 1 \cdot 1}{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1} \cdot \sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}}}{\sqrt[3]{\frac{2}{1 + e^{-2 \cdot x}} + 1}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{\color{blue}{e^{\mathsf{log1p}\left({\left(\sqrt[3]{1 + \frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}}\right)}^{2}\right)} - 1}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      Proof

      [Start] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \cdot \sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      expm1-log1p-u [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \cdot \sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right)\right)}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      expm1-udef [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \cdot \sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}\right)} - 1}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

   4. Simplified 100.0%

      \[\leadsto \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{\color{blue}{{\left(\sqrt[3]{1 + \frac{-2}{-1 - {\left(e^{x}\right)}^{-2}}}\right)}^{2}}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      Proof

      [Start] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{e^{\mathsf{log1p}\left({\left(\sqrt[3]{1 + \frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}}\right)}^{2}\right)} - 1}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      expm1-def [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{1 + \frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}}\right)}^{2}\right)\right)}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      expm1-log1p [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{\color{blue}{{\left(\sqrt[3]{1 + \frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}}\right)}^{2}}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      exp-prod [<=] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{-2}{-1 - \color{blue}{e^{-2 \cdot x}}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      *-commutative [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{-2}{-1 - e^{\color{blue}{x \cdot -2}}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      exp-prod [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{-2}{-1 - \color{blue}{{\left(e^{x}\right)}^{-2}}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\color{blue}{\left({\left(2 \cdot \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} + 1\right)}^{0.3333333333333333}\right)}}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\color{blue}{\left(\sqrt[3]{1 + \frac{2}{1 + e^{-\left(x + x\right)}}}\right)}}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      Proof

      [Start] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left({\left(2 \cdot \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} + 1\right)}^{0.3333333333333333}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      unpow1/3 [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\color{blue}{\left(\sqrt[3]{2 \cdot \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}} + 1}\right)}}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      +-commutative [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{\color{blue}{1 + 2 \cdot \frac{1}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      +-commutative [<=] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + 2 \cdot \frac{1}{\color{blue}{\frac{1}{{\left(e^{x}\right)}^{2}} + 1}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      associate-*r/ [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \color{blue}{\frac{2 \cdot 1}{\frac{1}{{\left(e^{x}\right)}^{2}} + 1}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      metadata-eval [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{\color{blue}{2}}{\frac{1}{{\left(e^{x}\right)}^{2}} + 1}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      +-commutative [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{2}{\color{blue}{1 + \frac{1}{{\left(e^{x}\right)}^{2}}}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      unpow2 [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{2}{1 + \frac{1}{\color{blue}{e^{x} \cdot e^{x}}}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      prod-exp [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{2}{1 + \frac{1}{\color{blue}{e^{x + x}}}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]
      rec-exp [=>] 100.0	\[ \frac{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{2}{1 + \color{blue}{e^{-\left(x + x\right)}}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto \frac{\frac{\color{blue}{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} - 1}}{{\left(\sqrt[3]{1 + \frac{2}{1 + e^{-\left(x + x\right)}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

   if -0.10000000000000001 < (*.f64 -2 x) < 2e-3

   1. Initial program 7.9%

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

   2. Simplified 7.9%

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

      [Start] 7.9	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      sub-neg [=>] 7.9	\[ \color{blue}{\frac{2}{1 + e^{-2 \cdot x}} + \left(-1\right)} \]
      *-commutative [=>] 7.9	\[ \frac{2}{1 + e^{\color{blue}{x \cdot -2}}} + \left(-1\right) \]
      exp-prod [=>] 7.9	\[ \frac{2}{1 + \color{blue}{{\left(e^{x}\right)}^{-2}}} + \left(-1\right) \]
      metadata-eval [=>] 7.9	\[ \frac{2}{1 + {\left(e^{x}\right)}^{-2}} + \color{blue}{-1} \]

   3. Taylor expanded in x around 0 100.0%

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

   if 2e-3 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 97.1%

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

   3. Simplified 97.1%

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

      [Start] 97.1	\[ \frac{2}{2 + -2 \cdot x} - 1 \]
      *-commutative [=>] 97.1	\[ \frac{2}{2 + \color{blue}{x \cdot -2}} - 1 \]

   4. Taylor expanded in x around inf 98.8%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{\frac{4 \cdot \frac{1}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} + -1}{{\left(\sqrt[3]{1 + \frac{2}{1 + e^{\left(-x\right) - x}}}\right)}^{2}}}{\sqrt[3]{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	26372

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;e^{\log \left(-1 + \frac{2}{1 + {\left(e^{x}\right)}^{-2}}\right)}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Accuracy	79.7%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.82:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 5
Accuracy	79.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 6
Accuracy	78.9%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 7
Accuracy	79.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 8
Accuracy	32.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 9
Accuracy	27.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))