Logistic function from Lakshay Garg

Percentage Accurate: 54.6% → 99.0%

Time: 7.2s

Alternatives: 6

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: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \left(x \cdot x\right)\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -10000000000.0)
   (+ (/ 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 (* x x))))
         0.3333333333333333))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -10000000000.0) {
		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 * (x * x)))) - 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) <= (-10000000000.0d0)) 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 * x)))) - 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) <= -10000000000.0) {
		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 * (x * x)))) - 0.3333333333333333)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -10000000000.0:
		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 * (x * x)))) - 0.3333333333333333)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -10000000000.0)
		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 * Float64(x * x)))) - 0.3333333333333333))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -10000000000.0)
		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 * x)))) - 0.3333333333333333)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -10000000000.0], 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[(x * x), $MachinePrecision]), $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) < -1e10

   1. Initial program 100.0%

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

   if -1e10 < (*.f64 #s(literal -2 binary64) x) < 0.050000000000000003

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot {x}^{2}\right) - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

   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 97.4%

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

      1. *-commutative 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.05:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.13333333333333333 + -0.05396825396825397 \cdot \left(x \cdot x\right)\right) - 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000000 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -10000000000.0) (not (<= (* -2.0 x) 5e-11)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (+
    x
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -10000000000.0) || !((-2.0 * x) <= 5e-11)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + (fma((x * x), 0.13333333333333333, -0.3333333333333333) * pow(x, 3.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -10000000000.0) || !(Float64(-2.0 * x) <= 5e-11))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x + Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * (x ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -10000000000.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-11]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x + N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -1e10 or 5.00000000000000018e-11 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

   if -1e10 < (*.f64 #s(literal -2 binary64) x) < 5.00000000000000018e-11

   1. Initial program 7.1%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.13333333333333333 \cdot {x}^{2} - 0.3333333333333333\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. fma-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. unpow2 100.0%

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

      9. unpow3 100.0%

         \[\leadsto x + \mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right) \cdot \color{blue}{{x}^{3}} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left({x}^{2}, 0.13333333333333333, -0.3333333333333333\right) \cdot {x}^{3}} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -10000000000 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot {x}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.005 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.005) (not (<= (* -2.0 x) 5e-11)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (* x (+ 1.0 (* (* x x) -0.3333333333333333)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.005) || !((-2.0 * x) <= 5e-11)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x * (1.0 + ((x * x) * -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) <= (-0.005d0)) .or. (.not. (((-2.0d0) * x) <= 5d-11))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x * (1.0d0 + ((x * x) * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.005) || !((-2.0 * x) <= 5e-11)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x * (1.0 + ((x * x) * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.005) or not ((-2.0 * x) <= 5e-11):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x * (1.0 + ((x * x) * -0.3333333333333333))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.005) || !(Float64(-2.0 * x) <= 5e-11))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.005) || ~(((-2.0 * x) <= 5e-11)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x * (1.0 + ((x * x) * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.005], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-11]], $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[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -0.0050000000000000001 or 5.00000000000000018e-11 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.9%

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

   if -0.0050000000000000001 < (*.f64 #s(literal -2 binary64) x) < 5.00000000000000018e-11

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. unpow3 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. unpow2 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.005 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.660000000000000031

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 6.6%

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

      1. +-commutative 6.6%

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

   5. Simplified 6.6%

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

      1. flip-- 6.5%

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

      2. metadata-eval 6.5%

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

      3. difference-of-sqr-1 6.5%

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

      4. associate-+l+ 6.5%

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

      5. metadata-eval 6.5%

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

      6. associate--l+ 69.5%

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

      7. metadata-eval 69.5%

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

      8. +-rgt-identity 69.5%

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

      9. associate-+l+ 69.5%

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

      10. metadata-eval 69.5%

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

   7. Applied egg-rr 69.5%

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

   8. Taylor expanded in x around 0 73.3%

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

      1. *-commutative 73.3%

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

   10. Simplified 73.3%

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

Alternative 5: 75.8% 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 97.4%

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

      1. *-commutative 97.4%

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

   5. Simplified 97.4%

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

   6. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 69.6%

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

Alternative 6: 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 53.5%

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

3. Taylor expanded in x around 0 29.7%

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

   1. *-commutative 29.7%

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

5. Simplified 29.7%

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

6. Taylor expanded in x around inf 29.1%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))