Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 98.8%

Time: 10.8s

Alternatives: 5

Speedup: 35.6×

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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \mathsf{expm1}\left(-2 \cdot x\right)\\ t_1 := {\left(e^{-2}\right)}^{x}\\ \mathbf{if}\;-2 \cdot x \leq -20000000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-1 + \frac{-8}{{\left(-1 - t_1\right)}^{3}}}}{\sqrt[3]{\frac{4}{{t_0}^{2}} + \left(1 + \frac{2}{t_0}\right)}} \cdot {\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t_1\right)\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 2.0 (expm1 (* -2.0 x)))) (t_1 (pow (exp -2.0) x)))
   (if (<= (* -2.0 x) -20000000000000.0)
     (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
     (if (<= (* -2.0 x) 4e-10)
       x
       (*
        (/
         (cbrt (+ -1.0 (/ -8.0 (pow (- -1.0 t_1) 3.0))))
         (cbrt (+ (/ 4.0 (pow t_0 2.0)) (+ 1.0 (/ 2.0 t_0)))))
        (pow (cbrt (expm1 (- (log 2.0) (log1p t_1)))) 2.0))))))

C

double code(double x, double y) {
	double t_0 = 2.0 + expm1((-2.0 * x));
	double t_1 = pow(exp(-2.0), x);
	double tmp;
	if ((-2.0 * x) <= -20000000000000.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 4e-10) {
		tmp = x;
	} else {
		tmp = (cbrt((-1.0 + (-8.0 / pow((-1.0 - t_1), 3.0)))) / cbrt(((4.0 / pow(t_0, 2.0)) + (1.0 + (2.0 / t_0))))) * pow(cbrt(expm1((log(2.0) - log1p(t_1)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 2.0 + Math.expm1((-2.0 * x));
	double t_1 = Math.pow(Math.exp(-2.0), x);
	double tmp;
	if ((-2.0 * x) <= -20000000000000.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 4e-10) {
		tmp = x;
	} else {
		tmp = (Math.cbrt((-1.0 + (-8.0 / Math.pow((-1.0 - t_1), 3.0)))) / Math.cbrt(((4.0 / Math.pow(t_0, 2.0)) + (1.0 + (2.0 / t_0))))) * Math.pow(Math.cbrt(Math.expm1((Math.log(2.0) - Math.log1p(t_1)))), 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 + expm1(Float64(-2.0 * x)))
	t_1 = exp(-2.0) ^ x
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20000000000000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-10)
		tmp = x;
	else
		tmp = Float64(Float64(cbrt(Float64(-1.0 + Float64(-8.0 / (Float64(-1.0 - t_1) ^ 3.0)))) / cbrt(Float64(Float64(4.0 / (t_0 ^ 2.0)) + Float64(1.0 + Float64(2.0 / t_0))))) * (cbrt(expm1(Float64(log(2.0) - log1p(t_1)))) ^ 2.0));
	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]}, Block[{t$95$1 = N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -20000000000000.0], 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], 4e-10], x, N[(N[(N[Power[N[(-1.0 + N[(-8.0 / N[Power[N[(-1.0 - t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -2e13 < (*.f64 -2 x) < 4.00000000000000015e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 4.00000000000000015e-10 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{2} \cdot \sqrt[3]{-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}}}{\sqrt[3]{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}} \cdot {\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{2}}}{\sqrt[3]{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)}} \]

      2. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}}}{\frac{\sqrt[3]{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)}}{{\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{2}}}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}}}{\sqrt[3]{1 + \left(\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + \frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}}\right)}} \cdot {\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{2}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}}}{\sqrt[3]{\frac{4}{{\left(2 + \mathsf{expm1}\left(x \cdot -2\right)\right)}^{2}} + \left(1 + \frac{2}{2 + \mathsf{expm1}\left(x \cdot -2\right)}\right)}} \cdot {\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20000000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-1 + \frac{-8}{{\left(-1 - {\left(e^{-2}\right)}^{x}\right)}^{3}}}}{\sqrt[3]{\frac{4}{{\left(2 + \mathsf{expm1}\left(-2 \cdot x\right)\right)}^{2}} + \left(1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\right)}} \cdot {\left(\sqrt[3]{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{2}\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20000000000000 \lor \neg \left(-2 \cdot x \leq 4 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -20000000000000.0) (not (<= (* -2.0 x) 4e-10)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   x))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -20000000000000.0) || !((-2.0 * x) <= 4e-10)) {
		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) <= (-20000000000000.0d0)) .or. (.not. (((-2.0d0) * x) <= 4d-10))) 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) <= -20000000000000.0) || !((-2.0 * x) <= 4e-10)) {
		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) <= -20000000000000.0) or not ((-2.0 * x) <= 4e-10):
		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) <= -20000000000000.0) || !(Float64(-2.0 * x) <= 4e-10))
		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) <= -20000000000000.0) || ~(((-2.0 * x) <= 4e-10)))
		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], -20000000000000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-10]], $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) < -2e13 or 4.00000000000000015e-10 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -2e13 < (*.f64 -2 x) < 4.00000000000000015e-10

   1. Initial program 6.6%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20000000000000 \lor \neg \left(-2 \cdot x \leq 4 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 78.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{x} \cdot \left(x + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.65) -1.0 (/ 1.0 (* (/ 0.5 x) (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.65) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((0.5 / x) * (x + 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 <= (-0.65d0)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 / ((0.5d0 / x) * (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.65) {
		tmp = -1.0;
	} else {
		tmp = 1.0 / ((0.5 / x) * (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.65:
		tmp = -1.0
	else:
		tmp = 1.0 / ((0.5 / x) * (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.65)
		tmp = -1.0;
	else
		tmp = Float64(1.0 / Float64(Float64(0.5 / x) * Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.65)
		tmp = -1.0;
	else
		tmp = 1.0 / ((0.5 / x) * (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.65], -1.0, N[(1.0 / N[(N[(0.5 / x), $MachinePrecision] * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.7%

      \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]
   3. Step-by-step derivation

      1. *-commutative 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.650000000000000022 < x

   1. Initial program 38.5%

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

   2. Taylor expanded in x around 0 6.1%

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

      1. +-commutative 6.1%

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

   4. Simplified 6.1%

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

      1. flip-- 6.0%

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

      2. clear-num 6.0%

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

      3. associate-+l+ 6.0%

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

      4. metadata-eval 6.0%

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

      5. metadata-eval 6.0%

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

      6. difference-of-sqr-1 6.0%

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

      7. associate-+l+ 6.0%

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

      8. metadata-eval 6.0%

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

      9. associate--l+ 67.2%

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

      10. metadata-eval 67.2%

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

      11. +-rgt-identity 67.2%

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

   6. Applied egg-rr 67.2%

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

      1. div-inv 67.2%

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

      2. *-commutative 67.2%

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

      3. +-commutative 67.2%

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

      4. associate-/r* 67.2%

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

      5. frac-2neg 67.2%

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

      6. metadata-eval 67.2%

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

      7. neg-sub0 67.2%

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

      8. +-commutative 67.2%

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

      9. associate--r+ 67.2%

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

      10. metadata-eval 67.2%

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

   8. Applied egg-rr 67.2%

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

   9. Taylor expanded in x around 0 71.7%

      \[\leadsto \frac{1}{\frac{\color{blue}{0.5}}{x} \cdot \left(2 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5}{x} \cdot \left(x + 2\right)}\\ \end{array} \]

Alternative 4: 75.7% accurate, 35.6× 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. Taylor expanded in x around 0 97.7%

      \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]
   3. Step-by-step derivation

      1. *-commutative 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 38.5%

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

   2. Taylor expanded in x around 0 67.5%

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

4. Final simplification 75.5%

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

Alternative 5: 27.3% 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 53.6%

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

2. Taylor expanded in x around 0 27.8%

   \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]
3. Step-by-step derivation

   1. *-commutative 27.8%

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

4. Simplified 27.8%

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

5. Taylor expanded in x around inf 26.9%

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

6. Final simplification 26.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))