Logistic function from Lakshay Garg

Percentage Accurate: 54.6% → 99.0%

Time: 9.0s

Alternatives: 9

Speedup: 9.9×

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: 54.6% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;\frac{2 \cdot \mathsf{expm1}\left(-2 \cdot x\right)}{\mathsf{expm1}\left(x \cdot -4\right)} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2.0)
   (+ (/ (* 2.0 (expm1 (* -2.0 x))) (expm1 (* x -4.0))) -1.0)
   (if (<= (* -2.0 x) 1e-8) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2.0) {
		tmp = ((2.0 * expm1((-2.0 * x))) / expm1((x * -4.0))) + -1.0;
	} else if ((-2.0 * x) <= 1e-8) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2.0) {
		tmp = ((2.0 * Math.expm1((-2.0 * x))) / Math.expm1((x * -4.0))) + -1.0;
	} else if ((-2.0 * x) <= 1e-8) {
		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) <= -2.0:
		tmp = ((2.0 * math.expm1((-2.0 * x))) / math.expm1((x * -4.0))) + -1.0
	elif (-2.0 * x) <= 1e-8:
		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) <= -2.0)
		tmp = Float64(Float64(Float64(2.0 * expm1(Float64(-2.0 * x))) / expm1(Float64(x * -4.0))) + -1.0);
	elseif (Float64(-2.0 * x) <= 1e-8)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2.0], N[(N[(N[(2.0 * N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(Exp[N[(x * -4.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-8], 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) < -2

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. exp-prod 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. flip-+ 99.9%

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

      3. pow-exp 99.9%

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

      4. *-commutative 99.9%

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

      5. expm1-udef 99.9%

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

      6. associate-/r/ 100.0%

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

      7. pow-sqr 100.0%

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

      8. metadata-eval 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. pow-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. expm1-def 100.0%

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

      8. *-commutative 100.0%

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

      9. associate-*r* 100.0%

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

      10. metadata-eval 100.0%

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

      11. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -2 < (*.f64 -2 x) < 1e-8

   1. Initial program 10.2%

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

      1. sub-neg 10.2%

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

      2. *-commutative 10.2%

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

      3. exp-prod 10.1%

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

      4. metadata-eval 10.1%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1e-8 < (*.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. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;\frac{2 \cdot \mathsf{expm1}\left(-2 \cdot x\right)}{\mathsf{expm1}\left(x \cdot -4\right)} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2.0)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 1e-8) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2.0) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 1e-8) {
		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) <= (-2.0d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 1d-8) 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) <= -2.0) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 1e-8) {
		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) <= -2.0:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 1e-8:
		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) <= -2.0)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 1e-8)
		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) <= -2.0)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 1e-8)
		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], -2.0], 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], 1e-8], 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) < -2

   1. Initial program 99.9%

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

   if -2 < (*.f64 -2 x) < 1e-8

   1. Initial program 10.2%

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

      1. sub-neg 10.2%

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

      2. *-commutative 10.2%

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

      3. exp-prod 10.1%

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

      4. metadata-eval 10.1%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1e-8 < (*.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. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 79.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15)
   -1.0
   (if (<= x 1.0)
     (+ x (* -0.3333333333333333 (pow x 3.0)))
     (* x (/ 2.0 (+ x 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15) {
		tmp = -1.0;
	} else if (x <= 1.0) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = x * (2.0 / (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 <= (-1.15d0)) then
        tmp = -1.0d0
    else if (x <= 1.0d0) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15) {
		tmp = -1.0;
	} else if (x <= 1.0) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15:
		tmp = -1.0
	elif x <= 1.0:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15)
		tmp = -1.0;
	elseif (x <= 1.0)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15)
		tmp = -1.0;
	elseif (x <= 1.0)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15], -1.0, If[LessEqual[x, 1.0], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   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. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -1.1499999999999999 < x < 1

   1. Initial program 10.2%

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

      1. sub-neg 10.2%

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

      2. *-commutative 10.2%

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

      3. exp-prod 10.1%

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

      4. metadata-eval 10.1%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. *-commutative 99.9%

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

      3. exp-prod 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 5.9%

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

      1. +-commutative 5.9%

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

   6. Simplified 5.9%

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

      1. flip-+ 5.5%

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

      2. metadata-eval 5.5%

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

      3. difference-of-sqr-1 5.5%

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

      4. associate-+l+ 5.5%

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

      5. metadata-eval 5.5%

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

      6. associate--l+ 5.5%

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

      7. metadata-eval 5.5%

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

      8. +-rgt-identity 5.5%

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

      9. associate--l+ 5.5%

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

      10. metadata-eval 5.5%

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

   8. Applied egg-rr 5.5%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

       1. +-commutative 18.8%

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

       2. *-un-lft-identity 18.8%

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

       3. times-frac 18.8%

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

       4. /-rgt-identity 18.8%

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

       5. +-commutative 18.8%

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

   13. Applied egg-rr 18.8%

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

4. Final simplification 77.8%

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

Alternative 4: 79.4% accurate, 7.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.5) x (- 2.0 (/ 4.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / 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 if (x <= 2.5d0) then
        tmp = x
    else
        tmp = 2.0d0 - (4.0d0 / 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 if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 2.5:
		tmp = x
	else:
		tmp = 2.0 - (4.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = Float64(2.0 - Float64(4.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = 2.0 - (4.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.5], x, N[(2.0 - N[(4.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 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. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2.5

   1. Initial program 10.9%

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

      1. sub-neg 10.9%

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

      2. *-commutative 10.9%

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

      3. exp-prod 10.8%

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

      4. metadata-eval 10.8%

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

   3. Simplified 10.8%

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

   4. Taylor expanded in x around 0 98.4%

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

   if 2.5 < 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. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

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

   6. Simplified 5.7%

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

      1. flip-+ 5.3%

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

      2. metadata-eval 5.3%

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

      3. difference-of-sqr-1 5.3%

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

      4. associate-+l+ 5.3%

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

      5. metadata-eval 5.3%

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

      6. associate--l+ 5.3%

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

      7. metadata-eval 5.3%

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

      8. +-rgt-identity 5.3%

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

      9. associate--l+ 5.3%

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

      10. metadata-eval 5.3%

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

   8. Applied egg-rr 5.3%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.8%

       \[\leadsto \color{blue}{2 - 4 \cdot \frac{1}{x}} \]
   13. Step-by-step derivation

       1. associate-*r/ 18.8%

          \[\leadsto 2 - \color{blue}{\frac{4 \cdot 1}{x}} \]

       2. metadata-eval 18.8%

          \[\leadsto 2 - \frac{\color{blue}{4}}{x} \]

   14. Simplified 18.8%

       \[\leadsto \color{blue}{2 - \frac{4}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

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

Alternative 5: 78.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66) -1.0 (* x (/ 2.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (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 <= (-0.66d0)) then
        tmp = -1.0d0
    else
        tmp = x * (2.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = x * (2.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = x * (2.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = Float64(x * Float64(2.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = x * (2.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(x * N[(2.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   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. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 42.7%

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

      1. sub-neg 42.7%

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

      2. *-commutative 42.7%

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

      3. exp-prod 42.7%

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

      4. metadata-eval 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in x around 0 8.4%

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

      1. +-commutative 8.4%

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

   6. Simplified 8.4%

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

      1. flip-+ 8.3%

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

      2. metadata-eval 8.3%

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

      3. difference-of-sqr-1 8.3%

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

      4. associate-+l+ 8.3%

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

      5. metadata-eval 8.3%

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

      6. associate--l+ 65.1%

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

      7. metadata-eval 65.1%

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

      8. +-rgt-identity 65.1%

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

      9. associate--l+ 65.1%

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

      10. metadata-eval 65.1%

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

   8. Applied egg-rr 65.1%

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

   9. Taylor expanded in x around 0 68.4%

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

       1. *-commutative 68.4%

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

   11. Simplified 68.4%

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

       1. +-commutative 68.4%

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

       2. *-un-lft-identity 68.4%

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

       3. times-frac 68.4%

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

       4. /-rgt-identity 68.4%

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

       5. +-commutative 68.4%

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

   13. Applied egg-rr 68.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 6: 78.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.66) -1.0 (/ (* x 2.0) (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) / (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 <= (-0.66d0)) then
        tmp = -1.0d0
    else
        tmp = (x * 2.0d0) / (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.66) {
		tmp = -1.0;
	} else {
		tmp = (x * 2.0) / (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = (x * 2.0) / (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = (x * 2.0) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   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. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 42.7%

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

      1. sub-neg 42.7%

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

      2. *-commutative 42.7%

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

      3. exp-prod 42.7%

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

      4. metadata-eval 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in x around 0 8.4%

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

      1. +-commutative 8.4%

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

   6. Simplified 8.4%

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

      1. flip-+ 8.3%

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

      2. metadata-eval 8.3%

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

      3. difference-of-sqr-1 8.3%

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

      4. associate-+l+ 8.3%

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

      5. metadata-eval 8.3%

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

      6. associate--l+ 65.1%

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

      7. metadata-eval 65.1%

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

      8. +-rgt-identity 65.1%

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

      9. associate--l+ 65.1%

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

      10. metadata-eval 65.1%

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

   8. Applied egg-rr 65.1%

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

   9. Taylor expanded in x around 0 68.4%

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

       1. *-commutative 68.4%

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

   11. Simplified 68.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \end{array} \]

Alternative 7: 79.4% accurate, 9.9× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.0) -1.0 (if (<= x 2.0) x 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else 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 <= (-1.0d0)) then
        tmp = -1.0d0
    else 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 <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.0) {
		tmp = x;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 2.0:
		tmp = x
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (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 <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.0)
		tmp = x;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.0], x, 2.0]]

Derivation

1. Split input into 3 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. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 10.9%

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

      1. sub-neg 10.9%

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

      2. *-commutative 10.9%

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

      3. exp-prod 10.8%

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

      4. metadata-eval 10.8%

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

   3. Simplified 10.8%

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

   4. Taylor expanded in x around 0 98.4%

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

   if 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. Taylor expanded in x around 0 5.7%

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

      1. +-commutative 5.7%

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

   6. Simplified 5.7%

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

      1. flip-+ 5.3%

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

      2. metadata-eval 5.3%

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

      3. difference-of-sqr-1 5.3%

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

      4. associate-+l+ 5.3%

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

      5. metadata-eval 5.3%

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

      6. associate--l+ 5.3%

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

      7. metadata-eval 5.3%

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

      8. +-rgt-identity 5.3%

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

      9. associate--l+ 5.3%

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

      10. metadata-eval 5.3%

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

   8. Applied egg-rr 5.3%

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

   9. Taylor expanded in x around 0 18.8%

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

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   12. Taylor expanded in x around inf 18.7%

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

4. Final simplification 77.3%

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

Alternative 8: 32.8% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.2e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.2e-308) {
		tmp = -1.0;
	} 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 <= 1.2d-308) then
        tmp = -1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.2e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.2e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.2e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.2e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.2e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1999999999999998e-308

   1. Initial program 55.4%

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

      1. sub-neg 55.4%

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

      2. *-commutative 55.4%

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

      3. exp-prod 55.4%

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

      4. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in x around 0 54.7%

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

      1. *-commutative 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in x around inf 54.3%

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

   if 1.1999999999999998e-308 < x

   1. Initial program 58.1%

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

      1. sub-neg 58.1%

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

      2. *-commutative 58.1%

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

      3. exp-prod 58.1%

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

      4. metadata-eval 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in x around 0 8.6%

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

      1. +-commutative 8.6%

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

   6. Simplified 8.6%

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

      1. flip-+ 8.4%

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

      2. metadata-eval 8.4%

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

      3. difference-of-sqr-1 8.4%

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

      4. associate-+l+ 8.4%

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

      5. metadata-eval 8.4%

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

      6. associate--l+ 49.7%

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

      7. metadata-eval 49.7%

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

      8. +-rgt-identity 49.7%

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

      9. associate--l+ 49.7%

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

      10. metadata-eval 49.7%

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

   8. Applied egg-rr 49.7%

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

   9. Taylor expanded in x around 0 55.2%

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

       1. *-commutative 55.2%

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

   11. Simplified 55.2%

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

   12. Taylor expanded in x around inf 12.7%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 9: 27.8% 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 56.8%

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

   1. sub-neg 56.8%

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

   2. *-commutative 56.8%

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

   3. exp-prod 56.8%

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

   4. metadata-eval 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in x around 0 29.0%

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

   1. *-commutative 29.0%

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

6. Simplified 29.0%

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

7. Taylor expanded in x around inf 26.8%

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

8. Final simplification 26.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))