Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 99.5%

Time: 6.7s

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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -200000.0)
   (- (/ 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 (* -0.05396825396825397 (pow x 2.0))))
         0.3333333333333333))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -200000.0) {
		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 + (-0.05396825396825397 * pow(x, 2.0)))) - 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) <= (-200000.0d0)) 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 + ((-0.05396825396825397d0) * (x ** 2.0d0)))) - 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) <= -200000.0) {
		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 + (-0.05396825396825397 * Math.pow(x, 2.0)))) - 0.3333333333333333)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -200000.0:
		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 + (-0.05396825396825397 * math.pow(x, 2.0)))) - 0.3333333333333333)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -200000.0)
		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(-0.05396825396825397 * (x ^ 2.0)))) - 0.3333333333333333))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -200000.0)
		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 + (-0.05396825396825397 * (x ^ 2.0)))) - 0.3333333333333333)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -200000.0], 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[(-0.05396825396825397 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -2e5

   1. Initial program 100.0%

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

   if -2e5 < (*.f64 -2 x) < 0.050000000000000003

   1. Initial program 9.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 -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 99.5%

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

   4. Taylor expanded in x around inf 100.0%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;-2 \cdot x \leq -200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   (if (<= (* -2.0 x) -200000.0)
     t_0
     (if (<= (* -2.0 x) 2e-7)
       (*
        x
        (+
         1.0
         (*
          (pow x 2.0)
          (- (* 0.13333333333333333 (pow x 2.0)) 0.3333333333333333))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	double tmp;
	if ((-2.0 * x) <= -200000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.13333333333333333 * pow(x, 2.0)) - 0.3333333333333333)));
	} else {
		tmp = t_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 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
    if (((-2.0d0) * x) <= (-200000.0d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 2d-7) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((0.13333333333333333d0 * (x ** 2.0d0)) - 0.3333333333333333d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
	double tmp;
	if ((-2.0 * x) <= -200000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.13333333333333333 * Math.pow(x, 2.0)) - 0.3333333333333333)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0
	tmp = 0
	if (-2.0 * x) <= -200000.0:
		tmp = t_0
	elif (-2.0 * x) <= 2e-7:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.13333333333333333 * math.pow(x, 2.0)) - 0.3333333333333333)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -200000.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 2e-7)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.13333333333333333 * (x ^ 2.0)) - 0.3333333333333333))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -200000.0)
		tmp = t_0;
	elseif ((-2.0 * x) <= 2e-7)
		tmp = x * (1.0 + ((x ^ 2.0) * ((0.13333333333333333 * (x ^ 2.0)) - 0.3333333333333333)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$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], -200000.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-7], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.13333333333333333 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -2e5 or 1.9999999999999999e-7 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -2e5 < (*.f64 -2 x) < 1.9999999999999999e-7

   1. Initial program 8.5%

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

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;-2 \cdot x \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   (if (<= (* -2.0 x) -0.002)
     t_0
     (if (<= (* -2.0 x) 2e-7) (+ (* -0.3333333333333333 (pow x 3.0)) x) t_0))))

C

double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	double tmp;
	if ((-2.0 * x) <= -0.002) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + x;
	} else {
		tmp = t_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 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
    if (((-2.0d0) * x) <= (-0.002d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 2d-7) then
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
	double tmp;
	if ((-2.0 * x) <= -0.002) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 2e-7) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0
	tmp = 0
	if (-2.0 * x) <= -0.002:
		tmp = t_0
	elif (-2.0 * x) <= 2e-7:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.002)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 2e-7)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -0.002)
		tmp = t_0;
	elseif ((-2.0 * x) <= 2e-7)
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$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], -0.002], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-7], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -2e-3 or 1.9999999999999999e-7 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -2e-3 < (*.f64 -2 x) < 1.9999999999999999e-7

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.3333333333333333 \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. associate-*l* 100.0%

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

      5. unpow2 100.0%

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

      6. pow3 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 \cdot x}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 + x \cdot \left(x \cdot \left(2 + -1.3333333333333333 \cdot x\right) - 2\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) 5e-13)
   (/ (* 2.0 x) (+ x 2.0))
   (-
    (/ 2.0 (+ 2.0 (* x (- (* x (+ 2.0 (* -1.3333333333333333 x))) 2.0))))
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 5e-13) {
		tmp = (2.0 * x) / (x + 2.0);
	} else {
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (-1.3333333333333333 * x))) - 2.0)))) - 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) <= 5d-13) then
        tmp = (2.0d0 * x) / (x + 2.0d0)
    else
        tmp = (2.0d0 / (2.0d0 + (x * ((x * (2.0d0 + ((-1.3333333333333333d0) * x))) - 2.0d0)))) - 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= 5e-13) {
		tmp = (2.0 * x) / (x + 2.0);
	} else {
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (-1.3333333333333333 * x))) - 2.0)))) - 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= 5e-13:
		tmp = (2.0 * x) / (x + 2.0)
	else:
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (-1.3333333333333333 * x))) - 2.0)))) - 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= 5e-13)
		tmp = Float64(Float64(2.0 * x) / Float64(x + 2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * Float64(2.0 + Float64(-1.3333333333333333 * x))) - 2.0)))) - 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= 5e-13)
		tmp = (2.0 * x) / (x + 2.0);
	else
		tmp = (2.0 / (2.0 + (x * ((x * (2.0 + (-1.3333333333333333 * x))) - 2.0)))) - 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-13], N[(N[(2.0 * x), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(2.0 + N[(x * N[(N[(x * N[(2.0 + N[(-1.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < 4.9999999999999999e-13

   1. Initial program 38.5%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip-- 6.7%

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

      2. metadata-eval 6.7%

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

      3. difference-of-sqr-1 6.7%

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

      4. +-commutative 6.7%

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

      5. associate-+l+ 6.7%

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

      6. metadata-eval 6.7%

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

      7. add-exp-log 6.7%

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

      8. expm1-define 6.7%

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

      9. log1p-define 67.9%

         \[\leadsto \frac{\left(x + 2\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right)}{\left(1 + x\right) + 1} \]

      10. expm1-log1p-u 67.9%

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

      11. +-commutative 67.9%

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

      12. associate-+l+ 67.9%

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

      13. metadata-eval 67.9%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in x around 0 71.9%

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

   if 4.9999999999999999e-13 < (*.f64 -2 x)

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 98.5%

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + -1.3333333333333333 \cdot x\right) - 2\right)}} - 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.2% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y):
	tmp = 0
	if x <= -0.66:
		tmp = -1.0
	else:
		tmp = x * (2.0 / (2.0 + x))
	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(2.0 + x)));
	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 / (2.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

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

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 39.0%

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

   3. Taylor expanded in x around 0 7.3%

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

      1. flip-- 7.2%

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

      2. metadata-eval 7.2%

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

      3. difference-of-sqr-1 7.2%

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

      4. +-commutative 7.2%

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

      5. associate-+l+ 7.2%

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

      6. metadata-eval 7.2%

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

      7. add-exp-log 7.2%

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

      8. expm1-define 7.2%

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

      9. log1p-define 67.9%

         \[\leadsto \frac{\left(x + 2\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right)}{\left(1 + x\right) + 1} \]

      10. expm1-log1p-u 67.9%

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

      11. +-commutative 67.9%

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

      12. associate-+l+ 67.9%

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

      13. metadata-eval 67.9%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in x around 0 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-/l* 71.6%

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

      3. +-commutative 71.6%

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

   8. Applied egg-rr 71.6%

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

Alternative 6: 78.2% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.66d0)) then
        tmp = -1.0d0
    else
        tmp = (2.0d0 * x) / (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 = (2.0 * x) / (x + 2.0);
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.66)
		tmp = -1.0;
	else
		tmp = Float64(Float64(2.0 * x) / 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 = (2.0 * x) / (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.66], -1.0, N[(N[(2.0 * x), $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. Add Preprocessing

   3. Taylor expanded in x around 0 99.5%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 39.0%

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

   3. Taylor expanded in x around 0 7.3%

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

      1. flip-- 7.2%

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

      2. metadata-eval 7.2%

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

      3. difference-of-sqr-1 7.2%

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

      4. +-commutative 7.2%

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

      5. associate-+l+ 7.2%

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

      6. metadata-eval 7.2%

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

      7. add-exp-log 7.2%

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

      8. expm1-define 7.2%

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

      9. log1p-define 67.9%

         \[\leadsto \frac{\left(x + 2\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right)}{\left(1 + x\right) + 1} \]

      10. expm1-log1p-u 67.9%

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

      11. +-commutative 67.9%

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

      12. associate-+l+ 67.9%

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

      13. metadata-eval 67.9%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in x around 0 71.6%

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

Alternative 7: 78.7% 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 99.5%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0 98.6%

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

   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 5.4%

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

      1. flip-- 5.1%

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

      2. metadata-eval 5.1%

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

      3. difference-of-sqr-1 5.1%

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

      4. +-commutative 5.1%

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

      5. associate-+l+ 5.1%

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

      6. metadata-eval 5.1%

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

      7. add-exp-log 5.1%

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

      8. expm1-define 5.1%

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

      9. log1p-define 5.1%

         \[\leadsto \frac{\left(x + 2\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right)}{\left(1 + x\right) + 1} \]

      10. expm1-log1p-u 5.1%

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

      11. +-commutative 5.1%

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

      12. associate-+l+ 5.1%

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

      13. metadata-eval 5.1%

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

   5. Applied egg-rr 5.1%

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

   6. Taylor expanded in x around 0 18.8%

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

   7. Taylor expanded in x around inf 18.8%

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

Alternative 8: 32.1% 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 56.6%

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

   3. Taylor expanded in x around 0 55.8%

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

   4. Taylor expanded in x around inf 54.7%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 51.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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. flip-- 6.8%

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

      2. metadata-eval 6.8%

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

      3. difference-of-sqr-1 6.8%

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

      4. +-commutative 6.8%

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

      5. associate-+l+ 6.8%

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

      6. metadata-eval 6.8%

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

      7. add-exp-log 6.8%

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

      8. expm1-define 6.8%

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

      9. log1p-define 55.4%

         \[\leadsto \frac{\left(x + 2\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x\right)}\right)}{\left(1 + x\right) + 1} \]

      10. expm1-log1p-u 55.4%

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

      11. +-commutative 55.4%

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

      12. associate-+l+ 55.4%

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

      13. metadata-eval 55.4%

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

   5. Applied egg-rr 55.4%

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

   6. Taylor expanded in x around 0 61.2%

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

   7. Taylor expanded in x around inf 11.5%

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

Alternative 9: 27.0% 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 53.5%

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

3. Taylor expanded in x around 0 28.3%

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

4. Taylor expanded in x around inf 26.1%

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

Reproduce

herbie shell --seed 2024050 -o generate:simplify
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))