Logistic function from Lakshay Garg

Percentage Accurate: 53.7% → 99.5%

Time: 12.5s

Alternatives: 8

Speedup: 21.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{\frac{4}{{t_0}^{2}} + -1}{1 + \frac{2}{t_0}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) -0.2)
     (/ (+ (/ 4.0 (pow t_0 2.0)) -1.0) (+ 1.0 (/ 2.0 t_0)))
     (if (<= (* -2.0 x) 5e-5)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       -1.0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = ((4.0 / pow(t_0, 2.0)) + -1.0) / (1.0 + (2.0 / t_0));
	} else if ((-2.0 * x) <= 5e-5) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + exp(((-2.0d0) * x))
    if (((-2.0d0) * x) <= (-0.2d0)) then
        tmp = ((4.0d0 / (t_0 ** 2.0d0)) + (-1.0d0)) / (1.0d0 + (2.0d0 / t_0))
    else if (((-2.0d0) * x) <= 5d-5) 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 t_0 = 1.0 + Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = ((4.0 / Math.pow(t_0, 2.0)) + -1.0) / (1.0 + (2.0 / t_0));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -0.2:
		tmp = ((4.0 / math.pow(t_0, 2.0)) + -1.0) / (1.0 + (2.0 / t_0))
	elif (-2.0 * x) <= 5e-5:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = Float64(Float64(Float64(4.0 / (t_0 ^ 2.0)) + -1.0) / Float64(1.0 + Float64(2.0 / t_0)));
	elseif (Float64(-2.0 * x) <= 5e-5)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + exp((-2.0 * x));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.2)
		tmp = ((4.0 / (t_0 ^ 2.0)) + -1.0) / (1.0 + (2.0 / t_0));
	elseif ((-2.0 * x) <= 5e-5)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[(N[(N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-5], 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) < -0.20000000000000001

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-commutative 100.0%

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

      5. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -0.20000000000000001 < (*.f64 -2 x) < 5.00000000000000024e-5

   1. Initial program 8.1%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000024e-5 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.4%

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

      1. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} + -1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.2)
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= (* -2.0 x) 5e-5) (+ x (* -0.3333333333333333 (pow x 3.0))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 5e-5) {
		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) <= (-0.2d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else if (((-2.0d0) * x) <= 5d-5) 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) <= -0.2) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else if ((-2.0 * x) <= 5e-5) {
		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) <= -0.2:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	elif (-2.0 * x) <= 5e-5:
		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) <= -0.2)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	elseif (Float64(-2.0 * x) <= 5e-5)
		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) <= -0.2)
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	elseif ((-2.0 * x) <= 5e-5)
		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], -0.2], 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], 5e-5], 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) < -0.20000000000000001

   1. Initial program 99.9%

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

   if -0.20000000000000001 < (*.f64 -2 x) < 5.00000000000000024e-5

   1. Initial program 8.1%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.00000000000000024e-5 < (*.f64 -2 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.4%

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

      1. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 80.2% accurate, 11.9× 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. Taylor expanded in x around 0 98.4%

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

      1. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2.5

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0 99.0%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   4. Simplified 5.4%

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

      1. flip-- 5.1%

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

      2. pow2 5.1%

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

      3. metadata-eval 5.1%

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

   6. Applied egg-rr 5.1%

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

      1. unpow2 5.1%

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

      2. difference-of-sqr-1 5.1%

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

      3. associate-+l+ 5.1%

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

      4. metadata-eval 5.1%

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

      5. *-rgt-identity 5.1%

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

      6. fma-neg 5.1%

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

      7. metadata-eval 5.1%

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

      8. fma-def 5.1%

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

      9. *-rgt-identity 5.1%

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

      10. associate-+l+ 5.1%

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

      11. metadata-eval 5.1%

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

      12. +-rgt-identity 5.1%

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

      13. associate-+l+ 5.1%

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

      14. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   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 80.1%

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

Alternative 4: 79.8% accurate, 12.0× 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. Taylor expanded in x around 0 98.4%

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

      1. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 37.7%

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

   2. Taylor expanded in x around 0 7.4%

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

      1. +-commutative 7.4%

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

   4. Simplified 7.4%

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

      1. flip-- 7.3%

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

      2. pow2 7.3%

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

      3. metadata-eval 7.3%

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

   6. Applied egg-rr 7.3%

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

      1. unpow2 7.3%

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

      2. difference-of-sqr-1 7.3%

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

      3. associate-+l+ 7.3%

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

      4. metadata-eval 7.3%

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

      5. *-rgt-identity 7.3%

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

      6. fma-neg 7.3%

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

      7. metadata-eval 7.3%

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

      8. fma-def 7.3%

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

      9. *-rgt-identity 7.3%

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

      10. associate-+l+ 69.1%

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

      11. metadata-eval 69.1%

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

      12. +-rgt-identity 69.1%

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

      13. associate-+l+ 69.1%

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

      14. metadata-eval 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in x around 0 72.6%

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

       1. *-commutative 72.6%

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

   11. Simplified 72.6%

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

       1. +-commutative 72.6%

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

       2. *-un-lft-identity 72.6%

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

       3. times-frac 72.6%

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

       4. rgt-mult-inverse 72.6%

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

       5. un-div-inv 72.6%

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

       6. associate-/l* 70.2%

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

       7. un-div-inv 70.2%

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

       8. associate-*l* 72.6%

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

       9. rgt-mult-inverse 72.6%

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

       10. *-commutative 72.6%

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

       11. *-un-lft-identity 72.6%

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

       12. +-commutative 72.6%

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

   13. Applied egg-rr 72.6%

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

4. Final simplification 79.5%

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

Alternative 5: 79.7% accurate, 12.0× 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. Taylor expanded in x around 0 98.4%

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

      1. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 37.7%

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

   2. Taylor expanded in x around 0 7.4%

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

      1. +-commutative 7.4%

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

   4. Simplified 7.4%

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

      1. flip-- 7.3%

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

      2. pow2 7.3%

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

      3. metadata-eval 7.3%

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

   6. Applied egg-rr 7.3%

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

      1. unpow2 7.3%

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

      2. difference-of-sqr-1 7.3%

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

      3. associate-+l+ 7.3%

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

      4. metadata-eval 7.3%

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

      5. *-rgt-identity 7.3%

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

      6. fma-neg 7.3%

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

      7. metadata-eval 7.3%

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

      8. fma-def 7.3%

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

      9. *-rgt-identity 7.3%

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

      10. associate-+l+ 69.1%

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

      11. metadata-eval 69.1%

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

      12. +-rgt-identity 69.1%

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

      13. associate-+l+ 69.1%

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

      14. metadata-eval 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in x around 0 72.6%

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

       1. *-commutative 72.6%

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

   11. Simplified 72.6%

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

4. Final simplification 79.5%

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

Alternative 6: 80.2% accurate, 21.3× 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. Taylor expanded in x around 0 98.4%

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

      1. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 8.7%

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

   2. Taylor expanded in x around 0 99.0%

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

   if 2 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   4. Simplified 5.4%

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

      1. flip-- 5.1%

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

      2. pow2 5.1%

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

      3. metadata-eval 5.1%

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

   6. Applied egg-rr 5.1%

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

      1. unpow2 5.1%

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

      2. difference-of-sqr-1 5.1%

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

      3. associate-+l+ 5.1%

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

      4. metadata-eval 5.1%

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

      5. *-rgt-identity 5.1%

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

      6. fma-neg 5.1%

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

      7. metadata-eval 5.1%

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

      8. fma-def 5.1%

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

      9. *-rgt-identity 5.1%

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

      10. associate-+l+ 5.1%

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

      11. metadata-eval 5.1%

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

      12. +-rgt-identity 5.1%

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

      13. associate-+l+ 5.1%

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

      14. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   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%

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

4. Final simplification 80.1%

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

Alternative 7: 33.0% accurate, 35.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.15e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e-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 <= 1.15d-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 <= 1.15e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.15e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.15e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999e-308

   1. Initial program 56.0%

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

   2. Taylor expanded in x around 0 54.7%

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

      1. *-commutative 54.7%

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

   4. Simplified 54.7%

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

   5. Taylor expanded in x around inf 54.4%

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

   if 1.1499999999999999e-308 < x

   1. Initial program 50.7%

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

   2. Taylor expanded in x around 0 6.5%

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

      1. +-commutative 6.5%

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

   4. Simplified 6.5%

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

      1. flip-- 6.4%

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

      2. pow2 6.4%

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

      3. metadata-eval 6.4%

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

   6. Applied egg-rr 6.4%

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

      1. unpow2 6.4%

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

      2. difference-of-sqr-1 6.4%

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

      3. associate-+l+ 6.4%

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

      4. metadata-eval 6.4%

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

      5. *-rgt-identity 6.4%

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

      6. fma-neg 6.4%

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

      7. metadata-eval 6.4%

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

      8. fma-def 6.4%

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

      9. *-rgt-identity 6.4%

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

      10. associate-+l+ 55.5%

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

      11. metadata-eval 55.5%

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

      12. +-rgt-identity 55.5%

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

      13. associate-+l+ 55.5%

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

      14. metadata-eval 55.5%

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

   8. Simplified 55.5%

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

   9. Taylor expanded in x around 0 61.3%

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

       1. *-commutative 61.3%

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

   11. Simplified 61.3%

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

   12. Taylor expanded in x around inf 11.5%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 8: 28.2% 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 53.3%

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

2. Taylor expanded in x around 0 28.8%

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

   1. *-commutative 28.8%

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

4. Simplified 28.8%

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

5. Taylor expanded in x around inf 27.3%

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

6. Final simplification 27.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))