Logistic function from Lakshay Garg

Percentage Accurate: 53.0% → 99.7%

Time: 11.5s

Alternatives: 7

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: 53.0% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1 \lor \neg \left(-2 \cdot x \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;2 \cdot \log \left(e^{\frac{1}{1 + {\left(e^{-2}\right)}^{x}} + -0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.1) (not (<= (* -2.0 x) 4e-8)))
   (* 2.0 (log (exp (+ (/ 1.0 (+ 1.0 (pow (exp -2.0) x))) -0.5))))
   x))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.1) || !((-2.0 * x) <= 4e-8)) {
		tmp = 2.0 * log(exp(((1.0 / (1.0 + pow(exp(-2.0), x))) + -0.5)));
	} 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) <= (-0.1d0)) .or. (.not. (((-2.0d0) * x) <= 4d-8))) then
        tmp = 2.0d0 * log(exp(((1.0d0 / (1.0d0 + (exp((-2.0d0)) ** x))) + (-0.5d0))))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.1) || !((-2.0 * x) <= 4e-8)) {
		tmp = 2.0 * Math.log(Math.exp(((1.0 / (1.0 + Math.pow(Math.exp(-2.0), x))) + -0.5)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.1) or not ((-2.0 * x) <= 4e-8):
		tmp = 2.0 * math.log(math.exp(((1.0 / (1.0 + math.pow(math.exp(-2.0), x))) + -0.5)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.1) || !(Float64(-2.0 * x) <= 4e-8))
		tmp = Float64(2.0 * log(exp(Float64(Float64(1.0 / Float64(1.0 + (exp(-2.0) ^ x))) + -0.5))));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.1) || ~(((-2.0 * x) <= 4e-8)))
		tmp = 2.0 * log(exp(((1.0 / (1.0 + (exp(-2.0) ^ x))) + -0.5)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-8]], $MachinePrecision]], N[(2.0 * N[Log[N[Exp[N[(N[(1.0 / N[(1.0 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.10000000000000001 or 4.0000000000000001e-8 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

      1. add-log-exp 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. log-prod 100.0%

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

      4. sub-neg 100.0%

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

      5. exp-prod 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. exp-prod 100.0%

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

      9. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. count-2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. exp-diff 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. exp-prod 100.0%

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

      5. div-exp 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. unpow1/2 100.0%

         \[\leadsto 2 \cdot \log \color{blue}{\left({\left(e^{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1}\right)}^{0.5}\right)} \]

      9. exp-prod 100.0%

         \[\leadsto 2 \cdot \log \color{blue}{\left(e^{\left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1\right) \cdot 0.5}\right)} \]

      10. *-commutative 100.0%

          \[\leadsto 2 \cdot \log \left(e^{\color{blue}{0.5 \cdot \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1\right)}}\right) \]

      11. distribute-rgt-in 100.0%

          \[\leadsto 2 \cdot \log \left(e^{\color{blue}{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} \cdot 0.5 + -1 \cdot 0.5}}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto 2 \cdot \log \left(e^{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} \cdot 0.5 + \color{blue}{-0.5}}\right) \]

      13. exp-sum 98.7%

          \[\leadsto 2 \cdot \log \color{blue}{\left(e^{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} \cdot 0.5} \cdot e^{-0.5}\right)} \]

   10. Simplified 100.0%

       \[\leadsto 2 \cdot \log \color{blue}{\left(e^{\frac{1}{1 + {\left(e^{-2}\right)}^{x}} + -0.5}\right)} \]

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 4.0000000000000001e-8

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1 \lor \neg \left(-2 \cdot x \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;2 \cdot \log \left(e^{\frac{1}{1 + {\left(e^{-2}\right)}^{x}} + -0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.1)
   (log (exp (+ (/ 2.0 (+ 1.0 (pow (exp -2.0) x))) -1.0)))
   (if (<= (* -2.0 x) 4e-8)
     x
     (+ (+ (+ 1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))) -1.0) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = log(exp(((2.0 / (1.0 + pow(exp(-2.0), x))) + -1.0)));
	} else if ((-2.0 * x) <= 4e-8) {
		tmp = x;
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + exp((-2.0 * x))))) + -1.0) + -1.0;
	}
	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) <= (-0.1d0)) then
        tmp = log(exp(((2.0d0 / (1.0d0 + (exp((-2.0d0)) ** x))) + (-1.0d0))))
    else if (((-2.0d0) * x) <= 4d-8) then
        tmp = x
    else
        tmp = ((1.0d0 + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))) + (-1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = Math.log(Math.exp(((2.0 / (1.0 + Math.pow(Math.exp(-2.0), x))) + -1.0)));
	} else if ((-2.0 * x) <= 4e-8) {
		tmp = x;
	} else {
		tmp = ((1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))))) + -1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.1:
		tmp = math.log(math.exp(((2.0 / (1.0 + math.pow(math.exp(-2.0), x))) + -1.0)))
	elif (-2.0 * x) <= 4e-8:
		tmp = x
	else:
		tmp = ((1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))) + -1.0) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = log(exp(Float64(Float64(2.0 / Float64(1.0 + (exp(-2.0) ^ x))) + -1.0)));
	elseif (Float64(-2.0 * x) <= 4e-8)
		tmp = x;
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))) + -1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.1)
		tmp = log(exp(((2.0 / (1.0 + (exp(-2.0) ^ x))) + -1.0)));
	elseif ((-2.0 * x) <= 4e-8)
		tmp = x;
	else
		tmp = ((1.0 + (2.0 / (1.0 + exp((-2.0 * x))))) + -1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[Log[N[Exp[N[(N[(2.0 / N[(1.0 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-8], x, N[(N[(N[(1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.10000000000000001

   1. Initial program 99.9%

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

      1. add-log-exp 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-prod 100.0%

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

      4. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 4.0000000000000001e-8

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 4.0000000000000001e-8 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} - 1 \]

      2. expm1-undefine 99.9%

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

      3. log1p-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;t\_0 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + t\_0\right) + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) -0.1)
     (+ t_0 -1.0)
     (if (<= (* -2.0 x) 4e-8) x (+ (+ (+ 1.0 t_0) -1.0) -1.0)))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = t_0 + -1.0;
	} else if ((-2.0 * x) <= 4e-8) {
		tmp = x;
	} else {
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))
    if (((-2.0d0) * x) <= (-0.1d0)) then
        tmp = t_0 + (-1.0d0)
    else if (((-2.0d0) * x) <= 4d-8) then
        tmp = x
    else
        tmp = ((1.0d0 + t_0) + (-1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + Math.exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = t_0 + -1.0;
	} else if ((-2.0 * x) <= 4e-8) {
		tmp = x;
	} else {
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 / (1.0 + math.exp((-2.0 * x)))
	tmp = 0
	if (-2.0 * x) <= -0.1:
		tmp = t_0 + -1.0
	elif (-2.0 * x) <= 4e-8:
		tmp = x
	else:
		tmp = ((1.0 + t_0) + -1.0) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = Float64(t_0 + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-8)
		tmp = x;
	else
		tmp = Float64(Float64(Float64(1.0 + t_0) + -1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.1)
		tmp = t_0 + -1.0;
	elseif ((-2.0 * x) <= 4e-8)
		tmp = x;
	else
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[(t$95$0 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-8], x, N[(N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.10000000000000001

   1. Initial program 99.9%

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

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 4.0000000000000001e-8

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 4.0000000000000001e-8 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} - 1 \]

      2. expm1-undefine 99.9%

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

      3. log1p-undefine 100.0%

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

      4. +-commutative 100.0%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1 \lor \neg \left(-2 \cdot x \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.1) (not (<= (* -2.0 x) 4e-8)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   x))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.1) || !((-2.0 * x) <= 4e-8)) {
		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) <= (-0.1d0)) .or. (.not. (((-2.0d0) * x) <= 4d-8))) 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) <= -0.1) || !((-2.0 * x) <= 4e-8)) {
		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) <= -0.1) or not ((-2.0 * x) <= 4e-8):
		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) <= -0.1) || !(Float64(-2.0 * x) <= 4e-8))
		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) <= -0.1) || ~(((-2.0 * x) <= 4e-8)))
		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], -0.1], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-8]], $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 #s(literal -2 binary64) x) < -0.10000000000000001 or 4.0000000000000001e-8 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -0.10000000000000001 < (*.f64 #s(literal -2 binary64) x) < 4.0000000000000001e-8

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1 \lor \neg \left(-2 \cdot x \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2\right) \cdot \frac{1}{x + 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * 2.0) * (1.0 / (x + 2.0))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (x * 2.0) * (1.0 / (x + 2.0));
end

Wolfram

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

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around 0 5.9%

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

   1. +-commutative 5.9%

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

5. Simplified 5.9%

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

   1. flip-- 5.7%

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

   2. div-inv 5.7%

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

   3. metadata-eval 5.7%

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

   4. difference-of-sqr-1 5.7%

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

   5. associate-+l+ 5.7%

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

   6. metadata-eval 5.7%

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

   7. associate--l+ 52.5%

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

   8. metadata-eval 52.5%

      \[\leadsto \left(\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)\right) \cdot \frac{1}{\left(x + 1\right) + 1} \]

   9. +-rgt-identity 52.5%

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

   10. associate-+l+ 52.5%

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

   11. metadata-eval 52.5%

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

7. Applied egg-rr 52.5%

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

8. Taylor expanded in x around 0 55.2%

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

   1. *-commutative 55.2%

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

10. Simplified 55.2%

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

11. Final simplification 55.2%

    \[\leadsto \left(x \cdot 2\right) \cdot \frac{1}{x + 2} \]
12. Add Preprocessing

Alternative 6: 56.7% accurate, 18.1× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= x 2.0) x 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.0) {
		tmp = x;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = x
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.0) {
		tmp = x;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.0:
		tmp = x
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.0)
		tmp = x;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = x;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.0], x, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 36.9%

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

   3. Taylor expanded in x around 0 69.2%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. +-commutative 5.1%

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

   5. Simplified 5.1%

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

      1. flip-- 4.7%

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

      2. div-inv 4.7%

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

      3. metadata-eval 4.7%

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

      4. difference-of-sqr-1 4.7%

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

      5. associate-+l+ 4.7%

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

      6. metadata-eval 4.7%

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

      7. associate--l+ 4.7%

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

      8. metadata-eval 4.7%

         \[\leadsto \left(\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)\right) \cdot \frac{1}{\left(x + 1\right) + 1} \]

      9. +-rgt-identity 4.7%

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

      10. associate-+l+ 4.7%

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

      11. metadata-eval 4.7%

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

   7. Applied egg-rr 4.7%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.8%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 6.9% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 2.0)

C

double code(double x, double y) {
	return 2.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0
end function

Java

public static double code(double x, double y) {
	return 2.0;
}

Python

def code(x, y):
	return 2.0

Julia

function code(x, y)
	return 2.0
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0;
end

Wolfram

code[x_, y_] := 2.0

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in x around 0 5.9%

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

   1. +-commutative 5.9%

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

5. Simplified 5.9%

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

   1. flip-- 5.7%

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

   2. div-inv 5.7%

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

   3. metadata-eval 5.7%

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

   4. difference-of-sqr-1 5.7%

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

   5. associate-+l+ 5.7%

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

   6. metadata-eval 5.7%

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

   7. associate--l+ 52.5%

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

   8. metadata-eval 52.5%

      \[\leadsto \left(\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)\right) \cdot \frac{1}{\left(x + 1\right) + 1} \]

   9. +-rgt-identity 52.5%

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

   10. associate-+l+ 52.5%

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

   11. metadata-eval 52.5%

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

7. Applied egg-rr 52.5%

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

8. Taylor expanded in x around 0 55.2%

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

   1. *-commutative 55.2%

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

10. Simplified 55.2%

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

11. Taylor expanded in x around inf 7.1%

    \[\leadsto \color{blue}{2} \]

12. Final simplification 7.1%

    \[\leadsto 2 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))