Logistic function from Lakshay Garg

Percentage Accurate: 54.7% → 99.4%

Time: 11.9s

Alternatives: 18

Speedup: 5.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% 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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := -1 - t\_0\\ \mathbf{if}\;-2 \cdot x \leq -2000:\\ \;\;\;\;\frac{2}{2 + \frac{x + x}{\left(\left(x + x\right) - \left(x + x\right)\right) - \left(x + x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-2 + \frac{4}{t\_1}}{t\_1}\right) \cdot \frac{{\left(\mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)}^{-3} + -1}{{\left(1 + \frac{1 + \frac{2}{1 + t\_0}}{-0.5 \cdot t\_1}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (- -1.0 t_0)))
   (if (<= (* -2.0 x) -2000.0)
     (+ (/ 2.0 (+ 2.0 (/ (+ x x) (- (- (+ x x) (+ x x)) (+ x x))))) -1.0)
     (if (<= (* -2.0 x) 5e-10)
       (fma -0.3333333333333333 (* x (* x x)) x)
       (*
        (+ 1.0 (/ (+ -2.0 (/ 4.0 t_1)) t_1))
        (/
         (+ (pow (fma 0.5 t_0 0.5) -3.0) -1.0)
         (pow (+ 1.0 (/ (+ 1.0 (/ 2.0 (+ 1.0 t_0))) (* -0.5 t_1))) 2.0)))))))

C

double code(double x, double y) {
	double t_0 = exp((-2.0 * x));
	double t_1 = -1.0 - t_0;
	double tmp;
	if ((-2.0 * x) <= -2000.0) {
		tmp = (2.0 / (2.0 + ((x + x) / (((x + x) - (x + x)) - (x + x))))) + -1.0;
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = (1.0 + ((-2.0 + (4.0 / t_1)) / t_1)) * ((pow(fma(0.5, t_0, 0.5), -3.0) + -1.0) / pow((1.0 + ((1.0 + (2.0 / (1.0 + t_0))) / (-0.5 * t_1))), 2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(-1.0 - t_0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -2000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(Float64(x + x) / Float64(Float64(Float64(x + x) - Float64(x + x)) - Float64(x + x))))) + -1.0);
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-2.0 + Float64(4.0 / t_1)) / t_1)) * Float64(Float64((fma(0.5, t_0, 0.5) ^ -3.0) + -1.0) / (Float64(1.0 + Float64(Float64(1.0 + Float64(2.0 / Float64(1.0 + t_0))) / Float64(-0.5 * t_1))) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2000.0], N[(N[(2.0 / N[(2.0 + N[(N[(x + x), $MachinePrecision] / N[(N[(N[(x + x), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 + N[(N[(-2.0 + N[(4.0 / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(0.5 * t$95$0 + 0.5), $MachinePrecision], -3.0], $MachinePrecision] + -1.0), $MachinePrecision] / N[Power[N[(1.0 + N[(N[(1.0 + N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 100.0%

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

   if -2e3 < (*.f64 #s(literal -2 binary64) 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

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 5.00000000000000031e-10 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. frac-sub N/A

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(\left(1 + e^{-2 \cdot x}\right) \cdot 0.5\right)}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{{\color{blue}{\left(\frac{1}{2} \cdot \left(1 + e^{-2 \cdot x}\right)\right)}}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{{\left(\frac{1}{2} \cdot \color{blue}{\left(e^{-2 \cdot x} + 1\right)}\right)}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{{\color{blue}{\left(e^{-2 \cdot x} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}\right)}}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{{\left(e^{-2 \cdot x} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}\right)}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(e^{-2 \cdot x}, \frac{1}{2}, \frac{1}{2}\right)\right)}}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{fma}\left(\color{blue}{e^{-2 \cdot x}}, \frac{1}{2}, \frac{1}{2}\right)\right)}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]

      6. *-commutative N/A

         \[\leadsto \frac{{\left(\mathsf{fma}\left(e^{\color{blue}{x \cdot -2}}, \frac{1}{2}, \frac{1}{2}\right)\right)}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]

      7. lower-*.f64 100.0

         \[\leadsto \frac{{\left(\mathsf{fma}\left(e^{\color{blue}{x \cdot -2}}, 0.5, 0.5\right)\right)}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{fma}\left(e^{x \cdot -2}, 0.5, 0.5\right)\right)}}^{-3} \cdot \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) - \mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right) \cdot 1}{{\left(\mathsf{fma}\left(4, {\left(1 + e^{-2 \cdot x}\right)}^{-2}, 1 - \frac{-2}{1 + e^{-2 \cdot x}}\right)\right)}^{2}} \]

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{1 + \frac{2}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot -0.5}}{1} \cdot \frac{{\left(\mathsf{fma}\left(0.5, e^{-2 \cdot x}, 0.5\right)\right)}^{-3} - 1}{{\left(1 - \frac{1 + \frac{2}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot -0.5}\right)}^{2}}} \]

   10. Taylor expanded in x around inf

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

       1. sub-neg N/A

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

       2. lower-+.f64 N/A

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

       3. associate-*r/ N/A

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

       4. distribute-neg-frac2 N/A

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

       5. lower-/.f64 N/A

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

   12. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\left(1 + \frac{-2 + \frac{4}{-1 - e^{x \cdot -2}}}{-1 - e^{x \cdot -2}}\right)} \cdot \frac{{\left(\mathsf{fma}\left(0.5, e^{-2 \cdot x}, 0.5\right)\right)}^{-3} - 1}{{\left(1 - \frac{1 + \frac{2}{1 + e^{-2 \cdot x}}}{\left(1 + e^{-2 \cdot x}\right) \cdot -0.5}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2000:\\ \;\;\;\;\frac{2}{2 + \frac{x + x}{\left(\left(x + x\right) - \left(x + x\right)\right) - \left(x + x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{-2 + \frac{4}{-1 - e^{-2 \cdot x}}}{-1 - e^{-2 \cdot x}}\right) \cdot \frac{{\left(\mathsf{fma}\left(0.5, e^{-2 \cdot x}, 0.5\right)\right)}^{-3} + -1}{{\left(1 + \frac{1 + \frac{2}{1 + e^{-2 \cdot x}}}{-0.5 \cdot \left(-1 - e^{-2 \cdot x}\right)}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2000:\\ \;\;\;\;\frac{2}{2 + \frac{x + x}{\left(\left(x + x\right) - \left(x + x\right)\right) - \left(x + x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2000.0)
   (+ (/ 2.0 (+ 2.0 (/ (+ x x) (- (- (+ x x) (+ x x)) (+ x x))))) -1.0)
   (if (<= (* -2.0 x) 5e-10)
     (fma -0.3333333333333333 (* x (* x x)) x)
     (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2000.0) {
		tmp = (2.0 / (2.0 + ((x + x) / (((x + x) - (x + x)) - (x + x))))) + -1.0;
	} else if ((-2.0 * x) <= 5e-10) {
		tmp = fma(-0.3333333333333333, (x * (x * x)), x);
	} else {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -2000.0)
		tmp = Float64(Float64(2.0 / Float64(2.0 + Float64(Float64(x + x) / Float64(Float64(Float64(x + x) - Float64(x + x)) - Float64(x + x))))) + -1.0);
	elseif (Float64(-2.0 * x) <= 5e-10)
		tmp = fma(-0.3333333333333333, Float64(x * Float64(x * x)), x);
	else
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2000.0], N[(N[(2.0 / N[(2.0 + N[(N[(x + x), $MachinePrecision] / N[(N[(N[(x + x), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision] - N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-10], N[(-0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal -2 binary64) x) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. lower--.f64 N/A

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

      4. count-2 N/A

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

      5. lower-+.f64 1.6

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

   5. Applied rewrites 1.6%

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

   7. Applied rewrites 99.9%

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

   if -2e3 < (*.f64 #s(literal -2 binary64) x) < 5.00000000000000031e-10

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 5.00000000000000031e-10 < (*.f64 #s(literal -2 binary64) x)

   1. Initial program 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2000:\\ \;\;\;\;\frac{2}{2 + \frac{x + x}{\left(\left(x + x\right) - \left(x + x\right)\right) - \left(x + x\right)}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))