Logistic function from Lakshay Garg

Percentage Accurate: 55.4% → 99.9%

Time: 7.5s

Alternatives: 8

Speedup: 18.1×

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: 55.4% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;{\left(\sqrt{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1}\right)}^{2}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.5)
   (pow (sqrt (+ (/ 2.0 (+ 1.0 (pow (exp -2.0) x))) -1.0)) 2.0)
   (if (<= (* -2.0 x) 0.005)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (- (* (pow x 2.0) 0.13333333333333333) 0.3333333333333333))))
     (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = pow(sqrt(((2.0 / (1.0 + pow(exp(-2.0), x))) + -1.0)), 2.0);
	} else if ((-2.0 * x) <= 0.005) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	} else {
		tmp = -1.0 + (2.0 / (1.0 + exp((-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 (((-2.0d0) * x) <= (-0.5d0)) then
        tmp = sqrt(((2.0d0 / (1.0d0 + (exp((-2.0d0)) ** x))) + (-1.0d0))) ** 2.0d0
    else if (((-2.0d0) * x) <= 0.005d0) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * (((x ** 2.0d0) * 0.13333333333333333d0) - 0.3333333333333333d0)))
    else
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = Math.pow(Math.sqrt(((2.0 / (1.0 + Math.pow(Math.exp(-2.0), x))) + -1.0)), 2.0);
	} else if ((-2.0 * x) <= 0.005) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	} else {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.5:
		tmp = math.pow(math.sqrt(((2.0 / (1.0 + math.pow(math.exp(-2.0), x))) + -1.0)), 2.0)
	elif (-2.0 * x) <= 0.005:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * 0.13333333333333333) - 0.3333333333333333)))
	else:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = sqrt(Float64(Float64(2.0 / Float64(1.0 + (exp(-2.0) ^ x))) + -1.0)) ^ 2.0;
	elseif (Float64(-2.0 * x) <= 0.005)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * 0.13333333333333333) - 0.3333333333333333))));
	else
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.5)
		tmp = sqrt(((2.0 / (1.0 + (exp(-2.0) ^ x))) + -1.0)) ^ 2.0;
	elseif ((-2.0 * x) <= 0.005)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * 0.13333333333333333) - 0.3333333333333333)));
	else
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[Power[N[Sqrt[N[(N[(2.0 / N[(1.0 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005], 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], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.5

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

      3. sub-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 0.0050000000000000001

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 0.0050000000000000001 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;{\left(\sqrt{\frac{2}{1 + {\left(e^{-2}\right)}^{x}} + -1}\right)}^{2}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5 \lor \neg \left(-2 \cdot x \leq 0.005\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.5) (not (<= (* -2.0 x) 0.005)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (*
    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) <= -0.5) || !((-2.0 * x) <= 0.005)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} 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) <= (-0.5d0)) .or. (.not. (((-2.0d0) * x) <= 0.005d0))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    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) <= -0.5) || !((-2.0 * x) <= 0.005)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} 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) <= -0.5) or not ((-2.0 * x) <= 0.005):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	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) <= -0.5) || !(Float64(-2.0 * x) <= 0.005))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	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) <= -0.5) || ~(((-2.0 * x) <= 0.005)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	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], -0.5], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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) < -0.5 or 0.0050000000000000001 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 0.0050000000000000001

   1. Initial program 8.4%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5 \lor \neg \left(-2 \cdot x \leq 0.005\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.13333333333333333 - 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5 \lor \neg \left(-2 \cdot x \leq 0.001\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.5) (not (<= (* -2.0 x) 0.001)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.5) || !((-2.0 * x) <= 0.001)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	}
	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) <= (-0.5d0)) .or. (.not. (((-2.0d0) * x) <= 0.001d0))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.5) || !((-2.0 * x) <= 0.001)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.5) or not ((-2.0 * x) <= 0.001):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.5) || !(Float64(-2.0 * x) <= 0.001))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.5) || ~(((-2.0 * x) <= 0.001)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.001]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.5 or 1e-3 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -0.5 < (*.f64 #s(literal -2 binary64) x) < 1e-3

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. associate-*l* 99.9%

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

      5. unpow2 99.9%

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

      6. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5 \lor \neg \left(-2 \cdot x \leq 0.001\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 2}}{\frac{0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.02e-8)
   (+
    -1.0
    (/ 2.0 (+ 2.0 (* x (- (* x (+ 2.0 (* x -1.3333333333333333))) 2.0)))))
   (/ (/ 1.0 (+ x 2.0)) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	} else {
		tmp = (1.0 / (x + 2.0)) / (0.5 / 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 <= (-1.02d-8)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + (x * ((x * (2.0d0 + (x * (-1.3333333333333333d0)))) - 2.0d0))))
    else
        tmp = (1.0d0 / (x + 2.0d0)) / (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	} else {
		tmp = (1.0 / (x + 2.0)) / (0.5 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.02e-8:
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))))
	else:
		tmp = (1.0 / (x + 2.0)) / (0.5 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.02e-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(1.0 / Float64(x + 2.0)) / Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.02e-8)
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * (2.0 + (x * -1.3333333333333333))) - 2.0))));
	else
		tmp = (1.0 / (x + 2.0)) / (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.02e-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[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02000000000000003e-8

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 96.7%

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

   if -1.02000000000000003e-8 < x

   1. Initial program 39.7%

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

   3. Taylor expanded in x around 0 6.1%

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

      1. +-commutative 6.1%

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

   5. Simplified 6.1%

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

      1. flip-- 6.0%

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

      2. div-inv 6.0%

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

      3. metadata-eval 6.0%

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

      4. difference-of-sqr-1 6.0%

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

      5. associate-+l+ 6.0%

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

      6. metadata-eval 6.0%

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

      7. associate--l+ 66.2%

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

      8. metadata-eval 66.2%

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

      9. +-rgt-identity 66.2%

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

      10. associate-+l+ 66.2%

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

      11. metadata-eval 66.2%

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

   7. Applied egg-rr 66.2%

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

      1. div-inv 66.2%

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

      2. clear-num 66.0%

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

      3. div-inv 66.0%

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

      4. associate-/r* 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/r* 66.0%

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

   9. Applied egg-rr 66.0%

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

   10. Taylor expanded in x around 0 70.6%

       \[\leadsto \frac{\frac{1}{x + 2}}{\color{blue}{\frac{0.5}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot \left(2 + x \cdot -1.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 2}}{\frac{0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot 2 - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 2}}{\frac{0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.02e-8)
   (+ -1.0 (/ 2.0 (+ 2.0 (* x (- (* x 2.0) 2.0)))))
   (/ (/ 1.0 (+ x 2.0)) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * 2.0) - 2.0))));
	} else {
		tmp = (1.0 / (x + 2.0)) / (0.5 / 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 <= (-1.02d-8)) then
        tmp = (-1.0d0) + (2.0d0 / (2.0d0 + (x * ((x * 2.0d0) - 2.0d0))))
    else
        tmp = (1.0d0 / (x + 2.0d0)) / (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-8) {
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * 2.0) - 2.0))));
	} else {
		tmp = (1.0 / (x + 2.0)) / (0.5 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.02e-8:
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * 2.0) - 2.0))))
	else:
		tmp = (1.0 / (x + 2.0)) / (0.5 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.02e-8)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(2.0 + Float64(x * Float64(Float64(x * 2.0) - 2.0)))));
	else
		tmp = Float64(Float64(1.0 / Float64(x + 2.0)) / Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.02e-8)
		tmp = -1.0 + (2.0 / (2.0 + (x * ((x * 2.0) - 2.0))));
	else
		tmp = (1.0 / (x + 2.0)) / (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.02e-8], N[(-1.0 + N[(2.0 / N[(2.0 + N[(x * N[(N[(x * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.02000000000000003e-8

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 95.7%

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

   if -1.02000000000000003e-8 < x

   1. Initial program 39.7%

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

   3. Taylor expanded in x around 0 6.1%

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

      1. +-commutative 6.1%

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

   5. Simplified 6.1%

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

      1. flip-- 6.0%

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

      2. div-inv 6.0%

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

      3. metadata-eval 6.0%

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

      4. difference-of-sqr-1 6.0%

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

      5. associate-+l+ 6.0%

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

      6. metadata-eval 6.0%

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

      7. associate--l+ 66.2%

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

      8. metadata-eval 66.2%

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

      9. +-rgt-identity 66.2%

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

      10. associate-+l+ 66.2%

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

      11. metadata-eval 66.2%

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

   7. Applied egg-rr 66.2%

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

      1. div-inv 66.2%

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

      2. clear-num 66.0%

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

      3. div-inv 66.0%

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

      4. associate-/r* 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-/r* 66.0%

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

   9. Applied egg-rr 66.0%

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

   10. Taylor expanded in x around 0 70.6%

       \[\leadsto \frac{\frac{1}{x + 2}}{\color{blue}{\frac{0.5}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-8}:\\ \;\;\;\;-1 + \frac{2}{2 + x \cdot \left(x \cdot 2 - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 2}}{\frac{0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.67:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 2}}{\frac{0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.67) -1.0 (/ (/ 1.0 (+ x 2.0)) (/ 0.5 x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.67) {
		tmp = -1.0;
	} else {
		tmp = (1.0 / (x + 2.0)) / (0.5 / 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 <= (-0.67d0)) then
        tmp = -1.0d0
    else
        tmp = (1.0d0 / (x + 2.0d0)) / (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.67) {
		tmp = -1.0;
	} else {
		tmp = (1.0 / (x + 2.0)) / (0.5 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.67:
		tmp = -1.0
	else:
		tmp = (1.0 / (x + 2.0)) / (0.5 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.67)
		tmp = -1.0;
	else
		tmp = Float64(Float64(1.0 / Float64(x + 2.0)) / Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.67)
		tmp = -1.0;
	else
		tmp = (1.0 / (x + 2.0)) / (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.67], -1.0, N[(N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.67000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.5%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -0.67000000000000004 < x

   1. Initial program 40.7%

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

   3. Taylor expanded in x around 0 7.1%

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

      1. +-commutative 7.1%

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

   5. Simplified 7.1%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   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+ 65.9%

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

      8. metadata-eval 65.9%

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

      9. +-rgt-identity 65.9%

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

      10. associate-+l+ 65.9%

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

      11. metadata-eval 65.9%

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

   7. Applied egg-rr 65.9%

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

      1. div-inv 65.9%

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

      2. clear-num 65.7%

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

      3. div-inv 65.7%

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

      4. associate-/r* 65.7%

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

      5. *-commutative 65.7%

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

      6. associate-/r* 65.7%

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

   9. Applied egg-rr 65.7%

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

   10. Taylor expanded in x around 0 69.9%

       \[\leadsto \frac{\frac{1}{x + 2}}{\color{blue}{\frac{0.5}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.0% accurate, 18.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.5%

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

   4. Taylor expanded in x around inf 98.9%

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

   if -1 < x

   1. Initial program 40.7%

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

   3. Taylor expanded in x around 0 66.0%

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

Alternative 8: 27.7% accurate, 109.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 56.0%

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

3. Taylor expanded in x around 0 29.7%

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

4. Taylor expanded in x around inf 28.0%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))