Logistic function from Lakshay Garg

Percentage Accurate: 54.4% → 99.7%

Time: 18.1s

Alternatives: 8

Speedup: 21.3×

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: 54.4% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + t_0} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-9}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\mathsf{log1p}\left(t_0 \cdot 0.5 + -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -20.0)
     (+ (/ 2.0 (+ 1.0 t_0)) -1.0)
     (if (<= (* -2.0 x) 2e-9)
       (+
        x
        (+
         (* -0.3333333333333333 (pow x 3.0))
         (* 0.13333333333333333 (pow x 5.0))))
       (expm1 (- (log1p (+ (* t_0 0.5) -0.5))))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -20.0) {
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	} else if ((-2.0 * x) <= 2e-9) {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = expm1(-log1p(((t_0 * 0.5) + -0.5)));
	}
	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) <= -20.0) {
		tmp = (2.0 / (1.0 + t_0)) + -1.0;
	} else if ((-2.0 * x) <= 2e-9) {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = Math.expm1(-Math.log1p(((t_0 * 0.5) + -0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -20.0:
		tmp = (2.0 / (1.0 + t_0)) + -1.0
	elif (-2.0 * x) <= 2e-9:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	else:
		tmp = math.expm1(-math.log1p(((t_0 * 0.5) + -0.5)))
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + t_0)) + -1.0);
	elseif (Float64(-2.0 * x) <= 2e-9)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = expm1(Float64(-log1p(Float64(Float64(t_0 * 0.5) + -0.5))));
	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], -20.0], N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-9], 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[(Exp[(-N[Log[1 + N[(N[(t$95$0 * 0.5), $MachinePrecision] + -0.5), $MachinePrecision]], $MachinePrecision])] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -20

   1. Initial program 100.0%

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

   if -20 < (*.f64 -2 x) < 2.00000000000000012e-9

   1. Initial program 8.8%

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

   2. 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 2.00000000000000012e-9 < (*.f64 -2 x)

   1. Initial program 99.8%

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

      1. add-log-exp 99.8%

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

      2. *-un-lft-identity 99.8%

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

      3. log-prod 99.8%

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

      4. metadata-eval 99.8%

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

      5. add-log-exp 99.8%

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

      6. add-exp-log 99.8%

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

      7. expm1-def 99.8%

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

      8. log-div 99.8%

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

      9. log1p-udef 99.9%

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

      10. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-lft-identity 99.9%

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

   5. Simplified 99.9%

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

      1. log1p-udef 99.8%

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

      2. diff-log 99.8%

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

      3. clear-num 99.8%

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

      4. log-rec 99.9%

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

      5. div-inv 99.9%

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

      6. metadata-eval 99.9%

         \[\leadsto \mathsf{expm1}\left(-\log \left(\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot \color{blue}{0.5}\right)\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot 0.5\right)}\right) \]
   8. Step-by-step derivation

      1. log1p-expm1-u 99.9%

         \[\leadsto \mathsf{expm1}\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot 0.5\right)\right)\right)}\right) \]

      2. expm1-udef 99.9%

         \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot 0.5\right)} - 1}\right)\right) \]

      3. add-exp-log 99.9%

         \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(\color{blue}{\left(1 + {\left(e^{-2}\right)}^{x}\right) \cdot 0.5} - 1\right)\right) \]

      4. *-commutative 99.9%

         \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(\color{blue}{0.5 \cdot \left(1 + {\left(e^{-2}\right)}^{x}\right)} - 1\right)\right) \]

      5. distribute-lft-in 99.9%

         \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot {\left(e^{-2}\right)}^{x}\right)} - 1\right)\right) \]

      6. metadata-eval 99.9%

         \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(\left(\color{blue}{0.5} + 0.5 \cdot {\left(e^{-2}\right)}^{x}\right) - 1\right)\right) \]

   9. Applied egg-rr 99.9%

      \[\leadsto \mathsf{expm1}\left(-\color{blue}{\mathsf{log1p}\left(\left(0.5 + 0.5 \cdot {\left(e^{-2}\right)}^{x}\right) - 1\right)}\right) \]
   10. Step-by-step derivation

       1. +-commutative 99.9%

          \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot {\left(e^{-2}\right)}^{x} + 0.5\right)} - 1\right)\right) \]

       2. associate--l+ 100.0%

          \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(\color{blue}{0.5 \cdot {\left(e^{-2}\right)}^{x} + \left(0.5 - 1\right)}\right)\right) \]

       3. exp-prod 100.0%

          \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(0.5 \cdot \color{blue}{e^{-2 \cdot x}} + \left(0.5 - 1\right)\right)\right) \]

       4. *-commutative 100.0%

          \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(0.5 \cdot e^{\color{blue}{x \cdot -2}} + \left(0.5 - 1\right)\right)\right) \]

       5. metadata-eval 100.0%

          \[\leadsto \mathsf{expm1}\left(-\mathsf{log1p}\left(0.5 \cdot e^{x \cdot -2} + \color{blue}{-0.5}\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \mathsf{expm1}\left(-\color{blue}{\mathsf{log1p}\left(0.5 \cdot e^{x \cdot -2} + -0.5\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-9}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(-\mathsf{log1p}\left(e^{-2 \cdot x} \cdot 0.5 + -0.5\right)\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -20.0) (not (<= (* -2.0 x) 2e-9)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (+
    x
    (+
     (* -0.3333333333333333 (pow x 3.0))
     (* 0.13333333333333333 (pow x 5.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -20.0) || !((-2.0 * x) <= 2e-9)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + ((-0.3333333333333333 * pow(x, 3.0)) + (0.13333333333333333 * pow(x, 5.0)));
	}
	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) <= (-20.0d0)) .or. (.not. (((-2.0d0) * x) <= 2d-9))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    else
        tmp = x + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (0.13333333333333333d0 * (x ** 5.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -20.0) || !((-2.0 * x) <= 2e-9)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (0.13333333333333333 * Math.pow(x, 5.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -20.0) or not ((-2.0 * x) <= 2e-9):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	else:
		tmp = x + ((-0.3333333333333333 * math.pow(x, 3.0)) + (0.13333333333333333 * math.pow(x, 5.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -20.0) || !(Float64(-2.0 * x) <= 2e-9))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	else
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(0.13333333333333333 * (x ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -20.0) || ~(((-2.0 * x) <= 2e-9)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	else
		tmp = x + ((-0.3333333333333333 * (x ^ 3.0)) + (0.13333333333333333 * (x ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -20.0], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-9]], $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[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -20 or 2.00000000000000012e-9 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -20 < (*.f64 -2 x) < 2.00000000000000012e-9

   1. Initial program 8.8%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot {x}^{3} + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.001) (not (<= (* -2.0 x) 2e-9)))
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (+ x (* -0.3333333333333333 (pow x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.001) || !((-2.0 * x) <= 2e-9)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} 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) <= (-0.001d0)) .or. (.not. (((-2.0d0) * x) <= 2d-9))) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) + (-1.0d0)
    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) <= -0.001) || !((-2.0 * x) <= 2e-9)) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) + -1.0;
	} else {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.001) or not ((-2.0 * x) <= 2e-9):
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) + -1.0
	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) <= -0.001) || !(Float64(-2.0 * x) <= 2e-9))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	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) <= -0.001) || ~(((-2.0 * x) <= 2e-9)))
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	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], -0.001], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-9]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e-3 or 2.00000000000000012e-9 < (*.f64 -2 x)

   1. Initial program 99.7%

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

   if -1e-3 < (*.f64 -2 x) < 2.00000000000000012e-9

   1. Initial program 7.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 4: 79.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) -1.0 (if (<= x 2.5) x (- 2.0 (/ 4.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -1.0d0
    else if (x <= 2.5d0) then
        tmp = x
    else
        tmp = 2.0d0 - (4.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = -1.0;
	} else if (x <= 2.5) {
		tmp = x;
	} else {
		tmp = 2.0 - (4.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = -1.0
	elif x <= 2.5:
		tmp = x
	else:
		tmp = 2.0 - (4.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = Float64(2.0 - Float64(4.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -1.0;
	elseif (x <= 2.5)
		tmp = x;
	else
		tmp = 2.0 - (4.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], -1.0, If[LessEqual[x, 2.5], x, N[(2.0 - N[(4.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      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%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2.5

   1. Initial program 10.1%

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

   2. Taylor expanded in x around 0 97.7%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   4. Simplified 5.4%

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

      1. sub-neg 5.4%

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

      2. metadata-eval 5.4%

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

      3. flip-+ 5.0%

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

      4. metadata-eval 5.0%

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

      5. difference-of-sqr-1 5.0%

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

      6. associate-+l+ 5.0%

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

      7. metadata-eval 5.0%

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

      8. associate--l+ 5.0%

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

      9. metadata-eval 5.0%

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

      10. +-rgt-identity 5.0%

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

   6. Applied egg-rr 5.0%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   10. Taylor expanded in x around inf 18.8%

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

       1. associate-*r/ 18.8%

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

       2. metadata-eval 18.8%

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

   12. Simplified 18.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \]

Alternative 5: 78.7% accurate, 12.0× speedup

Math

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

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(x * Float64(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[(x * N[(2.0 / N[(x + 2.0), $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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      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%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -0.660000000000000031 < x

   1. Initial program 38.3%

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

   2. Taylor expanded in x around 0 7.4%

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

      1. +-commutative 7.4%

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

   4. Simplified 7.4%

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

      1. sub-neg 7.4%

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

      2. metadata-eval 7.4%

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

      3. flip-+ 7.3%

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

      4. metadata-eval 7.3%

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

      5. difference-of-sqr-1 7.3%

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

      6. associate-+l+ 7.3%

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

      7. metadata-eval 7.3%

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

      8. associate--l+ 68.7%

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

      9. metadata-eval 68.7%

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

      10. +-rgt-identity 68.7%

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

   6. Applied egg-rr 68.7%

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

   7. Taylor expanded in x around 0 72.1%

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

      1. *-commutative 72.1%

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

   9. Simplified 72.1%

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

       1. expm1-log1p-u 72.1%

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

       2. expm1-udef 11.1%

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

       3. associate--l+ 11.1%

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

       4. metadata-eval 11.1%

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

       5. *-un-lft-identity 11.1%

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

       6. times-frac 11.1%

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

       7. /-rgt-identity 11.1%

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

   11. Applied egg-rr 11.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{x + 2}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 72.1%

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

       2. expm1-log1p 72.1%

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

   13. Simplified 72.1%

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

4. Final simplification 78.4%

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

Alternative 6: 79.1% accurate, 21.3× 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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. log1p-udef 100.0%

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

      4. +-commutative 100.0%

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

      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%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around inf 100.0%

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

   if -1 < x < 2

   1. Initial program 10.1%

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

   2. Taylor expanded in x around 0 97.7%

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

   if 2 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 5.4%

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

      1. +-commutative 5.4%

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

   4. Simplified 5.4%

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

      1. sub-neg 5.4%

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

      2. metadata-eval 5.4%

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

      3. flip-+ 5.0%

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

      4. metadata-eval 5.0%

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

      5. difference-of-sqr-1 5.0%

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

      6. associate-+l+ 5.0%

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

      7. metadata-eval 5.0%

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

      8. associate--l+ 5.0%

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

      9. metadata-eval 5.0%

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

      10. +-rgt-identity 5.0%

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

   6. Applied egg-rr 5.0%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   10. Taylor expanded in x around inf 18.7%

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

4. Final simplification 79.1%

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

Alternative 7: 32.5% accurate, 35.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.1d-308) then
        tmp = -1.0d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-308

   1. Initial program 51.9%

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

      1. expm1-log1p-u 51.9%

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

      2. expm1-udef 51.9%

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

      3. log1p-udef 51.9%

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

      4. +-commutative 51.9%

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

      5. add-exp-log 51.9%

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

      6. +-commutative 51.9%

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

      7. exp-prod 51.9%

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

   3. Applied egg-rr 51.9%

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

   4. Taylor expanded in x around 0 50.4%

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

      1. *-commutative 50.4%

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

   6. Simplified 50.4%

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

   7. Taylor expanded in x around inf 50.0%

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

   if 1.1000000000000001e-308 < x

   1. Initial program 52.6%

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

   2. Taylor expanded in x around 0 7.5%

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

      1. +-commutative 7.5%

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

   4. Simplified 7.5%

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

      1. sub-neg 7.5%

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

      2. metadata-eval 7.5%

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

      3. flip-+ 7.4%

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

      4. metadata-eval 7.4%

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

      5. difference-of-sqr-1 7.4%

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

      6. associate-+l+ 7.4%

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

      7. metadata-eval 7.4%

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

      8. associate--l+ 54.4%

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

      9. metadata-eval 54.4%

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

      10. +-rgt-identity 54.4%

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

   6. Applied egg-rr 54.4%

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

   7. Taylor expanded in x around 0 59.8%

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

      1. *-commutative 59.8%

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

   9. Simplified 59.8%

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

   10. Taylor expanded in x around inf 11.9%

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

4. Final simplification 30.2%

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

Alternative 8: 27.4% accurate, 109.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 52.2%

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

   1. expm1-log1p-u 51.4%

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

   2. expm1-udef 51.3%

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

   3. log1p-udef 51.6%

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

   4. +-commutative 51.6%

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

   5. add-exp-log 52.2%

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

   6. +-commutative 52.2%

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

   7. exp-prod 52.2%

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

3. Applied egg-rr 52.2%

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

4. Taylor expanded in x around 0 26.9%

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

   1. *-commutative 26.9%

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

6. Simplified 26.9%

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

7. Taylor expanded in x around inf 25.0%

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

8. Final simplification 25.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023309 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))