Logistic function from Lakshay Garg

Average Error: 29.1 → 0.2

Time: 5.3s

Precision: binary64

Cost: 33412

Math

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

↓

\[\begin{array}{l} t_0 := 1 + {\left(e^{x}\right)}^{-2}\\ \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{\frac{4}{{t_0}^{2}} + -1}{1 + \frac{2}{t_0}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow (exp x) -2.0))))
   (if (<= (* -2.0 x) -0.2)
     (/ (+ (/ 4.0 (pow t_0 2.0)) -1.0) (+ 1.0 (/ 2.0 t_0)))
     (if (<= (* -2.0 x) 5e-5)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       -1.0))))

C

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

↓

double code(double x, double y) {
	double t_0 = 1.0 + pow(exp(x), -2.0);
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = ((4.0 / pow(t_0, 2.0)) + -1.0) / (1.0 + (2.0 / t_0));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

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

↓

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 = 1.0d0 + (exp(x) ** (-2.0d0))
    if (((-2.0d0) * x) <= (-0.2d0)) then
        tmp = ((4.0d0 / (t_0 ** 2.0d0)) + (-1.0d0)) / (1.0d0 + (2.0d0 / t_0))
    else if (((-2.0d0) * x) <= 5d-5) then
        tmp = x + ((-0.3333333333333333d0) * (x ** 3.0d0))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.pow(Math.exp(x), -2.0);
	double tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = ((4.0 / Math.pow(t_0, 2.0)) + -1.0) / (1.0 + (2.0 / t_0));
	} else if ((-2.0 * x) <= 5e-5) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = 1.0 + math.pow(math.exp(x), -2.0)
	tmp = 0
	if (-2.0 * x) <= -0.2:
		tmp = ((4.0 / math.pow(t_0, 2.0)) + -1.0) / (1.0 + (2.0 / t_0))
	elif (-2.0 * x) <= 5e-5:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	else:
		tmp = -1.0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(1.0 + (exp(x) ^ -2.0))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = Float64(Float64(Float64(4.0 / (t_0 ^ 2.0)) + -1.0) / Float64(1.0 + Float64(2.0 / t_0)));
	elseif (Float64(-2.0 * x) <= 5e-5)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = 1.0 + (exp(x) ^ -2.0);
	tmp = 0.0;
	if ((-2.0 * x) <= -0.2)
		tmp = ((4.0 / (t_0 ^ 2.0)) + -1.0) / (1.0 + (2.0 / t_0));
	elseif ((-2.0 * x) <= 5e-5)
		tmp = x + (-0.3333333333333333 * (x ^ 3.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.2], N[(N[(N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-5], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.20000000000000001

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Applied egg-rr 0.0

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

   if -0.20000000000000001 < (*.f64 -2 x) < 5.00000000000000024e-5

   1. Initial program 59.2

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

   2. Taylor expanded in x around 0 0.1

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

   if 5.00000000000000024e-5 < (*.f64 -2 x)

   1. Initial program 0.1

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

   2. Taylor expanded in x around 0 1.9

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

   3. Taylor expanded in x around inf 0.8

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{\frac{4}{{\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{2}} + -1}{1 + \frac{2}{1 + {\left(e^{x}\right)}^{-2}}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	26180

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13572

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{2}{1 + {\left(e^{x}\right)}^{-2}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	0.2
Cost	7304

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	15.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Error	46.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022295 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))