Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 98.7%

Time: 7.6s

Alternatives: 4

Speedup: 18.1×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
end function

Java

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

Python

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
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]

Initial Program: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
end function

Java

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

Python

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
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]

Alternative 1: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \mathsf{expm1}\left(-2 \cdot x\right)\\ \mathbf{if}\;-2 \cdot x \leq -40000000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-4}{{t\_0}^{2}}}{-1 + \frac{-2}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 2.0 (expm1 (* -2.0 x)))))
   (if (<= (* -2.0 x) -40000000000000.0)
     (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
     (if (<= (* -2.0 x) 1e-11)
       x
       (/ (+ 1.0 (/ -4.0 (pow t_0 2.0))) (+ -1.0 (/ -2.0 t_0)))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + expm1((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -40000000000000.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 1e-11) {
		tmp = x;
	} else {
		tmp = (1.0 + (-4.0 / pow(t_0, 2.0))) / (-1.0 + (-2.0 / t_0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + Math.expm1((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -40000000000000.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 1e-11) {
		tmp = x;
	} else {
		tmp = (1.0 + (-4.0 / Math.pow(t_0, 2.0))) / (-1.0 + (-2.0 / t_0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 + math.expm1((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -40000000000000.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 1e-11:
		tmp = x
	else:
		tmp = (1.0 + (-4.0 / math.pow(t_0, 2.0))) / (-1.0 + (-2.0 / t_0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 + expm1(Float64(-2.0 * x)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -40000000000000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 1e-11)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 + Float64(-4.0 / (t_0 ^ 2.0))) / Float64(-1.0 + Float64(-2.0 / t_0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 + N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -40000000000000.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-11], x, N[(N[(1.0 + N[(-4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -4e13

   1. Initial program 100.0%

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

   if -4e13 < (*.f64 -2 x) < 9.99999999999999939e-12

   1. Initial program 6.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Step-by-step derivation

      1. sub-neg 6.0%

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

      2. *-commutative 6.0%

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

      3. exp-prod 6.0%

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

      4. metadata-eval 6.0%

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

   3. Simplified 6.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x} \]

   if 9.99999999999999939e-12 < (*.f64 -2 x)

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{2}{1 + {\left(e^{x}\right)}^{-2}} + -1} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-+ 100.0%

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

      2. pow-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. pow-exp 100.0%

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

      5. *-commutative 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow-exp 100.0%

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

      8. *-commutative 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1 + \frac{-4}{{\left(2 + \mathsf{expm1}\left(x \cdot -2\right)\right)}^{2}}}{-1 + \frac{-2}{2 + \mathsf{expm1}\left(x \cdot -2\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -40000000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-4}{{\left(2 + \mathsf{expm1}\left(-2 \cdot x\right)\right)}^{2}}}{-1 + \frac{-2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -40000000000000 \lor \neg \left(-2 \cdot x \leq 10^{-11}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -40000000000000.0) (not (<= (* -2.0 x) 1e-11)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   x))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -40000000000000.0) || !((-2.0 * x) <= 1e-11)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((((-2.0d0) * x) <= (-40000000000000.0d0)) .or. (.not. (((-2.0d0) * x) <= 1d-11))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -40000000000000.0) || !((-2.0 * x) <= 1e-11)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -40000000000000.0) or not ((-2.0 * x) <= 1e-11):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -40000000000000.0) || !(Float64(-2.0 * x) <= 1e-11))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -40000000000000.0) || ~(((-2.0 * x) <= 1e-11)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -40000000000000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-11]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -4e13 or 9.99999999999999939e-12 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   if -4e13 < (*.f64 -2 x) < 9.99999999999999939e-12

   1. Initial program 6.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Step-by-step derivation

      1. sub-neg 6.0%

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

      2. *-commutative 6.0%

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

      3. exp-prod 6.0%

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

      4. metadata-eval 6.0%

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

   3. Simplified 6.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -40000000000000 \lor \neg \left(-2 \cdot x \leq 10^{-11}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 18.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) -1.0 x))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, x]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 95.6%

      \[\leadsto \frac{2}{1 + \color{blue}{\left(1 + -2 \cdot x\right)}} + -1 \]
   6. Step-by-step derivation

      1. *-commutative 95.6%

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

   7. Simplified 95.6%

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

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1} \]

   if -1 < x

   1. Initial program 39.1%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Step-by-step derivation

      1. sub-neg 39.1%

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

      2. *-commutative 39.1%

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

      3. exp-prod 39.1%

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

      4. metadata-eval 39.1%

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

   3. Simplified 39.1%

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

   5. Taylor expanded in x around 0 66.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.9% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 54.8%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
2. Step-by-step derivation

   1. sub-neg 54.8%

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

   2. *-commutative 54.8%

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

   3. exp-prod 54.8%

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

   4. metadata-eval 54.8%

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

3. Simplified 54.8%

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

5. Taylor expanded in x around 0 28.0%

   \[\leadsto \frac{2}{1 + \color{blue}{\left(1 + -2 \cdot x\right)}} + -1 \]
6. Step-by-step derivation

   1. *-commutative 28.0%

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

7. Simplified 28.0%

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

8. Taylor expanded in x around inf 28.0%

   \[\leadsto \color{blue}{-1} \]

9. Final simplification 28.0%

   \[\leadsto -1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))