Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 99.9%

Time: 9.4s

Alternatives: 6

Speedup: 35.6×

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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t_0\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.004:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + t_0} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -0.5)
     (expm1 (- (log 2.0) (log1p t_0)))
     (if (<= (* -2.0 x) 0.004)
       (+
        x
        (+
         (* -0.3333333333333333 (pow x 3.0))
         (* 0.13333333333333333 (pow x 5.0))))
       (+ (/ 2.0 (+ 1.0 t_0)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = expm1((log(2.0) - log1p(t_0)));
	} else if ((-2.0 * x) <= 0.004) {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(t_0)));
	} else if ((-2.0 * x) <= 0.004) {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -0.5:
		tmp = math.expm1((math.log(2.0) - math.log1p(t_0)))
	elif (-2.0 * x) <= 0.004:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = (2.0 / (1.0 + t_0)) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = expm1(Float64(log(2.0) - log1p(t_0)));
	elseif (Float64(-2.0 * x) <= 0.004)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = Float64(Float64(2.0 / Float64(1.0 + t_0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.004], N[(x + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 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-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. add-exp-log 100.0%

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

      7. expm1-def 100.0%

         \[\leadsto 0 + \color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)} \]

      8. log-div 100.0%

         \[\leadsto 0 + \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

      9. log1p-udef 100.0%

         \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right) \]

      10. exp-prod 100.0%

          \[\leadsto 0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

      2. exp-prod 100.0%

         \[\leadsto \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot x}}\right)\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{\color{blue}{x \cdot -2}}\right)\right) \]

      4. exp-prod 100.0%

         \[\leadsto \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{x}\right)}^{-2}}\right)\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)} \]
   7. Step-by-step derivation

      1. pow-exp 100.0%

         \[\leadsto \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{e^{x \cdot -2}}\right)\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{e^{x \cdot -2}}\right)\right) \]

   if -0.5 < (*.f64 -2 x) < 0.0040000000000000001

   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%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]

   if 0.0040000000000000001 < (*.f64 -2 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.004:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\left(\left(1 + t_0\right) + -1\right) + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.004:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) -0.5)
     (+ (+ (+ 1.0 t_0) -1.0) -1.0)
     (if (<= (* -2.0 x) 0.004)
       (+
        x
        (+
         (* -0.3333333333333333 (pow x 3.0))
         (* 0.13333333333333333 (pow x 5.0))))
       (+ t_0 -1.0)))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	} else if ((-2.0 * x) <= 0.004) {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))
    if (((-2.0d0) * x) <= (-0.5d0)) then
        tmp = ((1.0d0 + t_0) + (-1.0d0)) + (-1.0d0)
    else if (((-2.0d0) * x) <= 0.004d0) then
        tmp = x + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (0.13333333333333333d0 * (x ** 5.0d0)))
    else
        tmp = t_0 + (-1.0d0)
    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)));
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	} else if ((-2.0 * x) <= 0.004) {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 / (1.0 + math.exp((-2.0 * x)))
	tmp = 0
	if (-2.0 * x) <= -0.5:
		tmp = ((1.0 + t_0) + -1.0) + -1.0
	elif (-2.0 * x) <= 0.004:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = t_0 + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = Float64(Float64(Float64(1.0 + t_0) + -1.0) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.004)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = Float64(t_0 + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.5)
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	elseif ((-2.0 * x) <= 0.004)
		tmp = x + ((-0.3333333333333333 * (x ^ 3.0)) + (0.13333333333333333 * (x ^ 5.0)));
	else
		tmp = t_0 + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[(N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.004], N[(x + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 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. expm1-log1p-u 96.4%

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

      2. expm1-udef 96.4%

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

      3. log1p-udef 97.5%

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

      4. +-commutative 97.5%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.5 < (*.f64 -2 x) < 0.0040000000000000001

   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%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]

   if 0.0040000000000000001 < (*.f64 -2 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\left(\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) + -1\right) + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.004:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.5 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.5) (not (<= (* -2.0 x) 5e-10)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   x))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.5) || !((-2.0 * x) <= 5e-10)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -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 ((((-2.0d0) * x) <= (-0.5d0)) .or. (.not. (((-2.0d0) * x) <= 5d-10))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x
    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) <= 5e-10)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.5) or not ((-2.0 * x) <= 5e-10):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.5) || !(Float64(-2.0 * x) <= 5e-10))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.5) || ~(((-2.0 * x) <= 5e-10)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x;
	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], 5e-10]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.5 or 5.00000000000000031e-10 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -0.5 < (*.f64 -2 x) < 5.00000000000000031e-10

   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%

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

4. Final simplification 100.0%

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

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -0.5:\\ \;\;\;\;\left(\left(1 + t_0\right) + -1\right) + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0 + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) -0.5)
     (+ (+ (+ 1.0 t_0) -1.0) -1.0)
     (if (<= (* -2.0 x) 5e-10) x (+ t_0 -1.0)))))

C

double code(double x, double y) {
	double t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x;
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))
    if (((-2.0d0) * x) <= (-0.5d0)) then
        tmp = ((1.0d0 + t_0) + (-1.0d0)) + (-1.0d0)
    else if (((-2.0d0) * x) <= 5d-10) then
        tmp = x
    else
        tmp = t_0 + (-1.0d0)
    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)));
	double tmp;
	if ((-2.0 * x) <= -0.5) {
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = x;
	} else {
		tmp = t_0 + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 / (1.0 + math.exp((-2.0 * x)))
	tmp = 0
	if (-2.0 * x) <= -0.5:
		tmp = ((1.0 + t_0) + -1.0) + -1.0
	elif (-2.0 * x) <= 5e-10:
		tmp = x
	else:
		tmp = t_0 + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.5)
		tmp = Float64(Float64(Float64(1.0 + t_0) + -1.0) + -1.0);
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = x;
	else
		tmp = Float64(t_0 + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.5)
		tmp = ((1.0 + t_0) + -1.0) + -1.0;
	elseif ((-2.0 * x) <= 5e-10)
		tmp = x;
	else
		tmp = t_0 + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.5], N[(N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], x, N[(t$95$0 + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 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. expm1-log1p-u 96.4%

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

      2. expm1-udef 96.4%

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

      3. log1p-udef 97.5%

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

      4. +-commutative 97.5%

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

      5. add-exp-log 100.0%

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

      6. +-commutative 100.0%

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

      7. exp-prod 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.5 < (*.f64 -2 x) < 5.00000000000000031e-10

   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%

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

   if 5.00000000000000031e-10 < (*.f64 -2 x)

   1. Initial program 99.9%

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

4. Final simplification 100.0%

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

Alternative 5: 75.9% accurate, 35.6× 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 98.4%

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

      1. *-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in x around inf 99.3%

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

   if -1 < x

   1. Initial program 43.6%

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

   3. Taylor expanded in x around 0 63.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. 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 59.7%

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

3. Taylor expanded in x around 0 31.8%

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

   1. *-commutative 31.8%

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

5. Simplified 31.8%

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

6. Taylor expanded in x around inf 30.4%

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

7. Final simplification 30.4%

   \[\leadsto -1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))