Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 99.0%

Time: 8.5s

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: 54.5% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + {x}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -100000000.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.05)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (*
          (pow x 2.0)
          (+ 0.13333333333333333 (* (pow x 2.0) -0.05396825396825397)))
         0.3333333333333333))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -100000000.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.05) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.13333333333333333 + (pow(x, 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	} 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) <= (-100000000.0d0)) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.05d0) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (((x ** 2.0d0) * (0.13333333333333333d0 + ((x ** 2.0d0) * (-0.05396825396825397d0)))) - 0.3333333333333333d0)))
    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) <= -100000000.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.05) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.13333333333333333 + (Math.pow(x, 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -100000000.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	elif (-2.0 * x) <= 0.05:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.13333333333333333 + (math.pow(x, 2.0) * -0.05396825396825397))) - 0.3333333333333333)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -100000000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.05)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.13333333333333333 + Float64((x ^ 2.0) * -0.05396825396825397))) - 0.3333333333333333))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -100000000.0)
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	elseif ((-2.0 * x) <= 0.05)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.13333333333333333 + ((x ^ 2.0) * -0.05396825396825397))) - 0.3333333333333333)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000000.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.05], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.13333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e8

   1. Initial program 100.0%

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

   if -1e8 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 8.7%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]

   if 0.050000000000000003 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.4%

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

   4. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + {x}^{2} \cdot -0.05396825396825397\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000000 \lor \neg \left(-2 \cdot x \leq 10^{-10}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -100000000.0) (not (<= (* -2.0 x) 1e-10)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (*
    x
    (+
     1.0
     (*
      (pow x 2.0)
      (- (* (pow x 2.0) 0.13333333333333333) 0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -100000000.0) || !((-2.0 * x) <= 1e-10)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	}
	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) <= (-100000000.0d0)) .or. (.not. (((-2.0d0) * x) <= 1d-10))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (((x ** 2.0d0) * 0.13333333333333333d0) - 0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -100000000.0) || !((-2.0 * x) <= 1e-10)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -100000000.0) or not ((-2.0 * x) <= 1e-10):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -100000000.0) || !(Float64(-2.0 * x) <= 1e-10))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * 0.13333333333333333) - 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -100000000.0) || ~(((-2.0 * x) <= 1e-10)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-10]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e8 or 1.00000000000000004e-10 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -1e8 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000004e-10

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000000 \lor \neg \left(-2 \cdot x \leq 10^{-10}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000000 \lor \neg \left(-2 \cdot x \leq 10^{-10}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -100000000.0) (not (<= (* -2.0 x) 1e-10)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (* x (+ 1.0 (* (pow x 2.0) -0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -100000000.0) || !((-2.0 * x) <= 1e-10)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x * (1.0 + (pow(x, 2.0) * -0.3333333333333333));
	}
	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) <= (-100000000.0d0)) .or. (.not. (((-2.0d0) * x) <= 1d-10))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -100000000.0) || !((-2.0 * x) <= 1e-10)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -100000000.0) or not ((-2.0 * x) <= 1e-10):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x * (1.0 + (math.pow(x, 2.0) * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -100000000.0) || !(Float64(-2.0 * x) <= 1e-10))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -100000000.0) || ~(((-2.0 * x) <= 1e-10)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x * (1.0 + ((x ^ 2.0) * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-10]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e8 or 1.00000000000000004e-10 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -1e8 < (*.f64 #s(literal -2 binary64) x) < 1.00000000000000004e-10

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000000 \lor \neg \left(-2 \cdot x \leq 10^{-10}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-8)
   (+
    -1.0
    (/ 2.0 (+ 2.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0)))))
   (* (* x 2.0) (/ 1.0 (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	} else {
		tmp = (x * 2.0) * (1.0 / (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 <= (-1d-8)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + (x * ((x * (2.0d0 + (x * (-1.3333333333333333d0)))) - 2.0d0))))
    else
        tmp = (x * 2.0d0) * (1.0d0 / (x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	} else {
		tmp = (x * 2.0) * (1.0 / (x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-8:
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))
	else:
		tmp = (x * 2.0) * (1.0 / (x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-8)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * Float64(2.0 + Float64(x * -1.3333333333333333))) - 2.0)))));
	else
		tmp = Float64(Float64(x * 2.0) * Float64(1.0 / Float64(x + 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1e-8)
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	else
		tmp = (x * 2.0) * (1.0 / (x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-8], N[(-1.0 + N[(2.0 / N[(2.0 + N[(x * N[(N[(x * N[(2.0 + N[(x * -1.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + -1.3333333333333333 \cdot x\right) - 2\right)}} - 1 \]

   if -1e-8 < x

   1. Initial program 40.1%

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

   3. Taylor expanded in x around 0 6.9%

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

      1. +-commutative 6.9%

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

   5. Simplified 6.9%

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

      1. flip-- 6.8%

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

      2. div-inv 6.8%

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

      3. metadata-eval 6.8%

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

      4. difference-of-sqr-1 6.8%

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

      5. associate-+l+ 6.8%

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

      6. metadata-eval 6.8%

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

      7. associate--l+ 66.5%

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

      8. metadata-eval 66.5%

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

      9. +-rgt-identity 66.5%

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

      10. associate-+l+ 66.5%

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

      11. metadata-eval 66.5%

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

   7. Applied egg-rr 66.5%

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

   8. Taylor expanded in x around 0 70.5%

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

      1. *-commutative 70.5%

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

   10. Simplified 70.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.55:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.55) 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 <= 2.55) {
		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 <= 2.55d0) 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 <= 2.55) {
		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 <= 2.55:
		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 <= 2.55)
		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 <= 2.55)
		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, 2.55], 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. Add Preprocessing

   3. Taylor expanded in x around 0 99.4%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2.5499999999999998

   1. Initial program 8.7%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 2.5499999999999998 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. +-commutative 5.1%

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

   5. Simplified 5.1%

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

      1. flip-- 4.8%

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

      2. div-inv 4.8%

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

      3. metadata-eval 4.8%

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

      4. difference-of-sqr-1 4.8%

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

      5. associate-+l+ 4.8%

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

      6. metadata-eval 4.8%

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

      7. associate--l+ 4.8%

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

      8. metadata-eval 4.8%

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

      9. +-rgt-identity 4.8%

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

      10. associate-+l+ 4.8%

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

      11. metadata-eval 4.8%

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

   7. Applied egg-rr 4.8%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

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

       2. metadata-eval 18.8%

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

   13. Simplified 18.8%

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

Alternative 6: 79.3% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.650000000000000022

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.4%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -0.650000000000000022 < x

   1. Initial program 40.3%

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

   3. Taylor expanded in x around 0 7.0%

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

      1. +-commutative 7.0%

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

   5. Simplified 7.0%

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

      1. flip-- 6.9%

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

      2. div-inv 6.9%

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

      3. metadata-eval 6.9%

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

      4. difference-of-sqr-1 6.9%

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

      5. associate-+l+ 6.9%

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

      6. metadata-eval 6.9%

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

      7. associate--l+ 66.4%

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

      8. metadata-eval 66.4%

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

      9. +-rgt-identity 66.4%

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

      10. associate-+l+ 66.4%

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

      11. metadata-eval 66.4%

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

   7. Applied egg-rr 66.4%

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

   8. Taylor expanded in x around 0 70.3%

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

      1. *-commutative 70.3%

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

   10. Simplified 70.3%

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

Alternative 7: 79.8% 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. Add Preprocessing

   3. Taylor expanded in x around 0 99.4%

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

   4. 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. Add Preprocessing

   3. 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. Add Preprocessing

   3. Taylor expanded in x around 0 5.1%

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

      1. +-commutative 5.1%

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

   5. Simplified 5.1%

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

      1. flip-- 4.8%

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

      2. div-inv 4.8%

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

      3. metadata-eval 4.8%

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

      4. difference-of-sqr-1 4.8%

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

      5. associate-+l+ 4.8%

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

      6. metadata-eval 4.8%

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

      7. associate--l+ 4.8%

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

      8. metadata-eval 4.8%

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

      9. +-rgt-identity 4.8%

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

      10. associate-+l+ 4.8%

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

      11. metadata-eval 4.8%

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

   7. Applied egg-rr 4.8%

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

   8. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   10. Simplified 18.8%

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

   11. Taylor expanded in x around inf 18.8%

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

Alternative 8: 33.2% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.1e-308) -1.0 2.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-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 <= 1.1d-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 <= 1.1e-308) {
		tmp = -1.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e-308:
		tmp = -1.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e-308)
		tmp = -1.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-308], -1.0, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-308

   1. Initial program 51.3%

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

   3. Taylor expanded in x around 0 50.5%

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

   4. Taylor expanded in x around inf 49.9%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 54.3%

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

   3. Taylor expanded in x around 0 7.0%

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

      1. +-commutative 7.0%

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

   5. Simplified 7.0%

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

      1. flip-- 6.8%

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

      2. div-inv 6.8%

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

      3. metadata-eval 6.8%

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

      4. difference-of-sqr-1 6.8%

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

      5. associate-+l+ 6.8%

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

      6. metadata-eval 6.8%

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

      7. associate--l+ 52.2%

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

      8. metadata-eval 52.2%

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

      9. +-rgt-identity 52.2%

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

      10. associate-+l+ 52.2%

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

      11. metadata-eval 52.2%

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

   7. Applied egg-rr 52.2%

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

   8. Taylor expanded in x around 0 58.3%

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

      1. *-commutative 58.3%

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

   10. Simplified 58.3%

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

   11. Taylor expanded in x around inf 11.9%

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

Alternative 9: 28.3% accurate, 109.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 52.9%

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

3. Taylor expanded in x around 0 25.5%

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

4. Taylor expanded in x around inf 23.5%

   \[\leadsto \color{blue}{-1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))