Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 99.9%

Time: 14.3s

Alternatives: 7

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: 54.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{1 + e^{x \cdot -6}}, e^{x \cdot -4} - \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + -4 \cdot {\left(1 + t_0\right)}^{-2}}{-1 + \frac{2}{-1 - t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -0.05)
     (fma
      (/ 2.0 (+ 1.0 (exp (* x -6.0))))
      (- (exp (* x -4.0)) (expm1 (* -2.0 x)))
      -1.0)
     (if (<= (* -2.0 x) 5e-8)
       (+ x (* -0.3333333333333333 (* x (* x x))))
       (/
        (+ 1.0 (* -4.0 (pow (+ 1.0 t_0) -2.0)))
        (+ -1.0 (/ 2.0 (- -1.0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = fma((2.0 / (1.0 + exp((x * -6.0)))), (exp((x * -4.0)) - expm1((-2.0 * x))), -1.0);
	} else if ((-2.0 * x) <= 5e-8) {
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	} else {
		tmp = (1.0 + (-4.0 * pow((1.0 + t_0), -2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = fma(Float64(2.0 / Float64(1.0 + exp(Float64(x * -6.0)))), Float64(exp(Float64(x * -4.0)) - expm1(Float64(-2.0 * x))), -1.0);
	elseif (Float64(-2.0 * x) <= 5e-8)
		tmp = Float64(x + Float64(-0.3333333333333333 * Float64(x * Float64(x * x))));
	else
		tmp = Float64(Float64(1.0 + Float64(-4.0 * (Float64(1.0 + t_0) ^ -2.0))) / Float64(-1.0 + Float64(2.0 / Float64(-1.0 - t_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], -0.05], N[(N[(2.0 / N[(1.0 + N[Exp[N[(x * -6.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * -4.0), $MachinePrecision]], $MachinePrecision] - N[(Exp[N[(-2.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-8], N[(x + N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-4.0 * N[Power[N[(1.0 + t$95$0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(2.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. flip3-+ 99.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}{1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)}}} - 1 \]

      2. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right)} - 1 \]

      3. fma-neg 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}, 1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right), -1\right)} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{1 + {\left(e^{x}\right)}^{-6}}, e^{-4 \cdot x} - \mathsf{expm1}\left(-2 \cdot x\right), -1\right)} \]
   4. Step-by-step derivation

      1. pow-exp 100.0%

         \[\leadsto \mathsf{fma}\left(\frac{2}{1 + \color{blue}{e^{x \cdot -6}}}, e^{-4 \cdot x} - \mathsf{expm1}\left(-2 \cdot x\right), -1\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{2}{1 + \color{blue}{e^{x \cdot -6}}}, e^{-4 \cdot x} - \mathsf{expm1}\left(-2 \cdot x\right), -1\right) \]

   if -0.050000000000000003 < (*.f64 -2 x) < 4.9999999999999998e-8

   1. Initial program 7.5%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
   3. Step-by-step derivation

      1. unpow3 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-2}, -1\right) \cdot \frac{-1}{-1 + \frac{2}{-1 - {\left(e^{-2}\right)}^{x}}}} \]

   5. Simplified 99.9%

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

      1. *-lft-identity 99.9%

         \[\leadsto \frac{1 + \color{blue}{1 \cdot \frac{-4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      2. div-inv 99.9%

         \[\leadsto \frac{1 + 1 \cdot \color{blue}{\left(-4 \cdot \frac{1}{{\left(1 + e^{x \cdot -2}\right)}^{2}}\right)}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      3. pow-flip 100.0%

         \[\leadsto \frac{1 + 1 \cdot \left(-4 \cdot \color{blue}{{\left(1 + e^{x \cdot -2}\right)}^{\left(-2\right)}}\right)}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{1 + 1 \cdot \left(-4 \cdot {\left(1 + e^{x \cdot -2}\right)}^{\color{blue}{-2}}\right)}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{1 + 1 \cdot \left(-4 \cdot {\left(1 + e^{\color{blue}{-2 \cdot x}}\right)}^{-2}\right)}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      6. exp-prod 100.0%

         \[\leadsto \frac{1 + 1 \cdot \left(-4 \cdot {\left(1 + \color{blue}{{\left(e^{-2}\right)}^{x}}\right)}^{-2}\right)}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

         \[\leadsto \frac{1 + \color{blue}{-4 \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-2}}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      2. exp-prod 100.0%

         \[\leadsto \frac{1 + -4 \cdot {\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{-2}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      3. *-commutative 100.0%

         \[\leadsto \frac{1 + -4 \cdot {\left(1 + e^{\color{blue}{x \cdot -2}}\right)}^{-2}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{1 + e^{x \cdot -6}}, e^{x \cdot -4} - \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + -4 \cdot {\left(1 + e^{-2 \cdot x}\right)}^{-2}}{-1 + \frac{2}{-1 - e^{-2 \cdot x}}}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1 + -4 \cdot {\left(1 + t_0\right)}^{-2}}{-1 + \frac{2}{-1 - t_0}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 5e-8)))
     (/ (+ 1.0 (* -4.0 (pow (+ 1.0 t_0) -2.0))) (+ -1.0 (/ 2.0 (- -1.0 t_0))))
     (+ x (* -0.3333333333333333 (* x (* x x)))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 5e-8)) {
		tmp = (1.0 + (-4.0 * pow((1.0 + t_0), -2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)));
	} else {
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	}
	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 = exp(((-2.0d0) * x))
    if ((((-2.0d0) * x) <= (-0.05d0)) .or. (.not. (((-2.0d0) * x) <= 5d-8))) then
        tmp = (1.0d0 + ((-4.0d0) * ((1.0d0 + t_0) ** (-2.0d0)))) / ((-1.0d0) + (2.0d0 / ((-1.0d0) - t_0)))
    else
        tmp = x + ((-0.3333333333333333d0) * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.exp((-2.0 * x));
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 5e-8)) {
		tmp = (1.0 + (-4.0 * Math.pow((1.0 + t_0), -2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)));
	} else {
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.exp((-2.0 * x))
	tmp = 0
	if ((-2.0 * x) <= -0.05) or not ((-2.0 * x) <= 5e-8):
		tmp = (1.0 + (-4.0 * math.pow((1.0 + t_0), -2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)))
	else:
		tmp = x + (-0.3333333333333333 * (x * (x * x)))
	return tmp

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.05) || !(Float64(-2.0 * x) <= 5e-8))
		tmp = Float64(Float64(1.0 + Float64(-4.0 * (Float64(1.0 + t_0) ^ -2.0))) / Float64(-1.0 + Float64(2.0 / Float64(-1.0 - t_0))));
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = exp((-2.0 * x));
	tmp = 0.0;
	if (((-2.0 * x) <= -0.05) || ~(((-2.0 * x) <= 5e-8)))
		tmp = (1.0 + (-4.0 * ((1.0 + t_0) ^ -2.0))) / (-1.0 + (2.0 / (-1.0 - t_0)));
	else
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-8]], $MachinePrecision]], N[(N[(1.0 + N[(-4.0 * N[Power[N[(1.0 + t$95$0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + N[(2.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.050000000000000003 or 4.9999999999999998e-8 < (*.f64 -2 x)

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-2}, -1\right) \cdot \frac{-1}{-1 + \frac{2}{-1 - {\left(e^{-2}\right)}^{x}}}} \]

   5. Simplified 99.9%

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

      1. *-lft-identity 99.9%

         \[\leadsto \frac{1 + \color{blue}{1 \cdot \frac{-4}{{\left(1 + e^{x \cdot -2}\right)}^{2}}}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      2. div-inv 99.9%

         \[\leadsto \frac{1 + 1 \cdot \color{blue}{\left(-4 \cdot \frac{1}{{\left(1 + e^{x \cdot -2}\right)}^{2}}\right)}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      3. pow-flip 100.0%

         \[\leadsto \frac{1 + 1 \cdot \left(-4 \cdot \color{blue}{{\left(1 + e^{x \cdot -2}\right)}^{\left(-2\right)}}\right)}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{1 + 1 \cdot \left(-4 \cdot {\left(1 + e^{x \cdot -2}\right)}^{\color{blue}{-2}}\right)}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      5. *-commutative 100.0%

         \[\leadsto \frac{1 + 1 \cdot \left(-4 \cdot {\left(1 + e^{\color{blue}{-2 \cdot x}}\right)}^{-2}\right)}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      6. exp-prod 100.0%

         \[\leadsto \frac{1 + 1 \cdot \left(-4 \cdot {\left(1 + \color{blue}{{\left(e^{-2}\right)}^{x}}\right)}^{-2}\right)}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

         \[\leadsto \frac{1 + \color{blue}{-4 \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{-2}}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      2. exp-prod 100.0%

         \[\leadsto \frac{1 + -4 \cdot {\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{-2}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

      3. *-commutative 100.0%

         \[\leadsto \frac{1 + -4 \cdot {\left(1 + e^{\color{blue}{x \cdot -2}}\right)}^{-2}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

   9. Simplified 100.0%

      \[\leadsto \frac{1 + \color{blue}{-4 \cdot {\left(1 + e^{x \cdot -2}\right)}^{-2}}}{-1 + \frac{2}{-1 - e^{x \cdot -2}}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 4.9999999999999998e-8

   1. Initial program 7.5%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
   3. Step-by-step derivation

      1. unpow3 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1 + -4 \cdot {\left(1 + e^{-2 \cdot x}\right)}^{-2}}{-1 + \frac{2}{-1 - e^{-2 \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* -2.0 x) -0.05) (not (<= (* -2.0 x) 5e-8)))
   (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x)))))
   (+ x (* -0.3333333333333333 (* x (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 5e-8)) {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * (x * (x * 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) <= (-0.05d0)) .or. (.not. (((-2.0d0) * x) <= 5d-8))) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    else
        tmp = x + ((-0.3333333333333333d0) * (x * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((-2.0 * x) <= -0.05) || !((-2.0 * x) <= 5e-8)) {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	} else {
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((-2.0 * x) <= -0.05) or not ((-2.0 * x) <= 5e-8):
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	else:
		tmp = x + (-0.3333333333333333 * (x * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(-2.0 * x) <= -0.05) || !(Float64(-2.0 * x) <= 5e-8))
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	else
		tmp = Float64(x + Float64(-0.3333333333333333 * Float64(x * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((-2.0 * x) <= -0.05) || ~(((-2.0 * x) <= 5e-8)))
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	else
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Not[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-8]], $MachinePrecision]], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -0.050000000000000003 or 4.9999999999999998e-8 < (*.f64 -2 x)

   1. Initial program 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 4.9999999999999998e-8

   1. Initial program 7.5%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
   3. Step-by-step derivation

      1. unpow3 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;-1 + \frac{2}{1 + \frac{1}{e^{x + x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.05)
   (+ -1.0 (/ 2.0 (+ 1.0 (/ 1.0 (exp (+ x x))))))
   (if (<= (* -2.0 x) 5e-8)
     (+ x (* -0.3333333333333333 (* x (* x x))))
     (+ -1.0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = -1.0 + (2.0 / (1.0 + (1.0 / exp((x + x)))));
	} else if ((-2.0 * x) <= 5e-8) {
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	} else {
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((-2.0d0) * x) <= (-0.05d0)) then
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + (1.0d0 / exp((x + x)))))
    else if (((-2.0d0) * x) <= 5d-8) then
        tmp = x + ((-0.3333333333333333d0) * (x * (x * x)))
    else
        tmp = (-1.0d0) + (2.0d0 / (1.0d0 + exp(((-2.0d0) * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = -1.0 + (2.0 / (1.0 + (1.0 / Math.exp((x + x)))));
	} else if ((-2.0 * x) <= 5e-8) {
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	} else {
		tmp = -1.0 + (2.0 / (1.0 + Math.exp((-2.0 * x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = -1.0 + (2.0 / (1.0 + (1.0 / math.exp((x + x)))))
	elif (-2.0 * x) <= 5e-8:
		tmp = x + (-0.3333333333333333 * (x * (x * x)))
	else:
		tmp = -1.0 + (2.0 / (1.0 + math.exp((-2.0 * x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + Float64(1.0 / exp(Float64(x + x))))));
	elseif (Float64(-2.0 * x) <= 5e-8)
		tmp = Float64(x + Float64(-0.3333333333333333 * Float64(x * Float64(x * x))));
	else
		tmp = Float64(-1.0 + Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((-2.0 * x) <= -0.05)
		tmp = -1.0 + (2.0 / (1.0 + (1.0 / exp((x + x)))));
	elseif ((-2.0 * x) <= 5e-8)
		tmp = x + (-0.3333333333333333 * (x * (x * x)));
	else
		tmp = -1.0 + (2.0 / (1.0 + exp((-2.0 * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(-1.0 + N[(2.0 / N[(1.0 + N[(1.0 / N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-8], N[(x + N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. +-lft-identity 99.9%

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

      2. flip-+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. sub0-neg 99.9%

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

      5. pow2 99.9%

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

      6. exp-prod 99.9%

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

      7. associate--r+ 99.9%

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

      8. metadata-eval 99.9%

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

      9. exp-prod 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. exp-prod 99.9%

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

      2. *-commutative 99.9%

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

      3. exp-prod 99.9%

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

      4. *-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. frac-2neg 99.9%

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

      3. neg-sub0 99.9%

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

      4. metadata-eval 99.9%

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

      5. unpow2 99.9%

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

      6. neg-sub0 99.9%

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

      7. sub-neg 99.9%

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

      8. neg-sub0 99.9%

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

      9. associate--r- 99.9%

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

      10. metadata-eval 99.9%

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

      11. flip-- 99.9%

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

      12. neg-sub0 99.9%

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

      13. remove-double-neg 99.9%

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

      14. +-commutative 99.9%

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

      15. exp-prod 99.9%

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

      16. sqr-pow 99.9%

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

      17. fma-def 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\frac{1}{e^{x}}, \frac{1}{e^{x}}, 1\right)}} - 1 \]
   8. Step-by-step derivation

      1. rec-exp 99.9%

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

      2. rec-exp 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around inf 99.9%

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

       1. pow-exp 99.9%

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

       2. distribute-lft-neg-out 99.9%

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

       3. exp-neg 99.9%

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

       4. *-commutative 99.9%

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

       5. count-2 99.9%

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

   12. Applied egg-rr 99.9%

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

   if -0.050000000000000003 < (*.f64 -2 x) < 4.9999999999999998e-8

   1. Initial program 7.5%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x + -0.3333333333333333 \cdot {x}^{3}} \]
   3. Step-by-step derivation

      1. unpow3 100.0%

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

   4. Applied egg-rr 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 -2 x)

   1. Initial program 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;-1 + \frac{2}{1 + \frac{1}{e^{x + x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \end{array} \]

Alternative 5: 78.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.67:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x + x\right) \cdot \frac{1}{x + 2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.67000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.0%

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

      1. *-commutative 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -0.67000000000000004 < x

   1. Initial program 32.6%

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

   2. Taylor expanded in x around 0 5.6%

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

      1. *-commutative 5.6%

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

   4. Simplified 5.6%

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

   5. Taylor expanded in x around 0 7.1%

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

      1. +-commutative 7.1%

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

   7. Simplified 7.1%

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

      1. flip-- 7.0%

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

      2. div-inv 7.0%

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

      3. *-commutative 7.0%

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

      4. associate-+l+ 7.0%

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

      5. metadata-eval 7.0%

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

      6. metadata-eval 7.0%

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

      7. difference-of-sqr-1 7.0%

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

      8. associate--l+ 74.0%

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

      9. metadata-eval 74.0%

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

      10. +-rgt-identity 74.0%

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

      11. *-commutative 74.0%

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

      12. associate-+l+ 74.0%

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

      13. metadata-eval 74.0%

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

   9. Applied egg-rr 74.0%

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

   10. Taylor expanded in x around 0 76.9%

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

       1. count-2 76.9%

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

   12. Simplified 76.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.67:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x + x\right) \cdot \frac{1}{x + 2}\\ \end{array} \]

Alternative 6: 75.7% 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. Taylor expanded in x around 0 97.0%

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

      1. *-commutative 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -1 < x

   1. Initial program 32.6%

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

   2. Taylor expanded in x around 0 74.1%

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

4. Final simplification 81.4%

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

Alternative 7: 27.2% 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 51.5%

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

2. Taylor expanded in x around 0 31.3%

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

   1. *-commutative 31.3%

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

4. Simplified 31.3%

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

5. Taylor expanded in x around inf 30.4%

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

6. Final simplification 30.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))