Logistic function from Lakshay Garg

Percentage Accurate: 54.9% → 99.2%

Time: 5.5s

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: 54.9% 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: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2 \cdot \left(1 - e^{-2 \cdot x}\right)}{-\mathsf{expm1}\left(x \cdot -4\right)} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (+ (/ (* 2.0 (- 1.0 (exp (* -2.0 x)))) (- (expm1 (* x -4.0)))) -1.0)
   (if (<= (* -2.0 x) 4e-11)
     (+
      x
      (+
       (* -0.3333333333333333 (pow x 3.0))
       (* 0.13333333333333333 (pow x 5.0))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = ((2.0 * (1.0 - exp((-2.0 * x)))) / -expm1((x * -4.0))) + -1.0;
	} else if ((-2.0 * x) <= 4e-11) {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = ((2.0 * (1.0 - Math.exp((-2.0 * x)))) / -Math.expm1((x * -4.0))) + -1.0;
	} else if ((-2.0 * x) <= 4e-11) {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = ((2.0 * (1.0 - math.exp((-2.0 * x)))) / -math.expm1((x * -4.0))) + -1.0
	elif (-2.0 * x) <= 4e-11:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(Float64(Float64(2.0 * Float64(1.0 - exp(Float64(-2.0 * x)))) / Float64(-expm1(Float64(x * -4.0)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 4e-11)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(N[(N[(2.0 * N[(1.0 - N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-11], N[(x + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + 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) < -5

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. pow2 100.0%

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

      5. *-commutative 100.0%

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

      6. exp-prod 100.0%

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

      7. pow-pow 100.0%

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

      8. metadata-eval 100.0%

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

      9. exp-prod 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. exp-prod 100.0%

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

      3. *-commutative 100.0%

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

      4. exp-prod 100.0%

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

   5. Simplified 100.0%

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

      1. pow-exp 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. pow-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. exp-prod 100.0%

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

       3. sub-neg 100.0%

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

       4. +-commutative 100.0%

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

       5. neg-sub0 100.0%

          \[\leadsto \frac{2 \cdot \left(1 - e^{x \cdot -2}\right)}{\color{blue}{\left(0 - {\left(e^{x}\right)}^{-4}\right)} + 1} - 1 \]

       6. associate-+l- 100.0%

          \[\leadsto \frac{2 \cdot \left(1 - e^{x \cdot -2}\right)}{\color{blue}{0 - \left({\left(e^{x}\right)}^{-4} - 1\right)}} - 1 \]

       7. sub0-neg 100.0%

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

       8. exp-prod 100.0%

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

       9. *-commutative 100.0%

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

       10. expm1-def 100.0%

           \[\leadsto \frac{2 \cdot \left(1 - e^{x \cdot -2}\right)}{-\color{blue}{\mathsf{expm1}\left(-4 \cdot x\right)}} - 1 \]

       11. *-commutative 100.0%

           \[\leadsto \frac{2 \cdot \left(1 - e^{x \cdot -2}\right)}{-\mathsf{expm1}\left(\color{blue}{x \cdot -4}\right)} - 1 \]

   12. Simplified 100.0%

       \[\leadsto \frac{2 \cdot \left(1 - e^{x \cdot -2}\right)}{\color{blue}{-\mathsf{expm1}\left(x \cdot -4\right)}} - 1 \]

   if -5 < (*.f64 -2 x) < 3.99999999999999976e-11

   1. Initial program 7.5%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 3.99999999999999976e-11 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;\frac{2 \cdot \left(1 - e^{-2 \cdot x}\right)}{-\mathsf{expm1}\left(x \cdot -4\right)} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -5.0)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 4e-11)
     (+
      x
      (+
       (* -0.3333333333333333 (pow x 3.0))
       (* 0.13333333333333333 (pow x 5.0))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 4e-11) {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = -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) <= (-5.0d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 4d-11) then
        tmp = x + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (0.13333333333333333d0 * (x ** 5.0d0)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -5.0) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 4e-11) {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -5.0:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 4e-11:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -5.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 4e-11)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -5.0)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 4e-11)
		tmp = x + ((-0.3333333333333333 * (x ^ 3.0)) + (0.13333333333333333 * (x ^ 5.0)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -5.0], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-11], N[(x + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + 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) < -5

   1. Initial program 100.0%

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

   if -5 < (*.f64 -2 x) < 3.99999999999999976e-11

   1. Initial program 7.5%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 3.99999999999999976e-11 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -5:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.001)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 4e-11) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 4e-11) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -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) <= (-0.001d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 4d-11) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.001) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 4e-11) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.001:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 4e-11:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.001)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 4e-11)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.001)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 4e-11)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.001], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-11], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -1e-3

   1. Initial program 99.8%

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

   if -1e-3 < (*.f64 -2 x) < 3.99999999999999976e-11

   1. Initial program 6.9%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]

   if 3.99999999999999976e-11 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 75.3% 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 0 100.0%

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

      1. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 38.2%

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

   2. Taylor expanded in x around 0 68.6%

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

4. Final simplification 75.6%

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

Alternative 5: 27.4% 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 51.9%

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

2. Taylor expanded in x around 0 26.3%

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

   1. *-commutative 26.3%

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

4. Simplified 26.3%

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

5. Taylor expanded in x around inf 24.7%

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

6. Final simplification 24.7%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023321 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))