Logistic function from Lakshay Garg

Average Error: 28.9 → 0.3

Time: 17.5s

Precision: binary64

Cost: 7496

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;-2 \cdot x \leq -500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.3333333333333333 \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   (if (<= (* -2.0 x) -500.0)
     t_0
     (if (<= (* -2.0 x) 0.0002)
       (+ (* (* x x) (* -0.3333333333333333 x)) x)
       t_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 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	double tmp;
	if ((-2.0 * x) <= -500.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = ((x * x) * (-0.3333333333333333 * x)) + x;
	} else {
		tmp = t_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 = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
    if (((-2.0d0) * x) <= (-500.0d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 0.0002d0) then
        tmp = ((x * x) * ((-0.3333333333333333d0) * x)) + x
    else
        tmp = t_0
    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 = (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
	double tmp;
	if ((-2.0 * x) <= -500.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = ((x * x) * (-0.3333333333333333 * x)) + x;
	} else {
		tmp = t_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 = (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0
	tmp = 0
	if (-2.0 * x) <= -500.0:
		tmp = t_0
	elif (-2.0 * x) <= 0.0002:
		tmp = ((x * x) * (-0.3333333333333333 * x)) + x
	else:
		tmp = t_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(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -500.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = Float64(Float64(Float64(x * x) * Float64(-0.3333333333333333 * x)) + x);
	else
		tmp = t_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 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -500.0)
		tmp = t_0;
	elseif ((-2.0 * x) <= 0.0002)
		tmp = ((x * x) * (-0.3333333333333333 * x)) + x;
	else
		tmp = t_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[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -500.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -500 or 2.0000000000000001e-4 < (*.f64 -2 x)

   1. Initial program 0.0

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

   if -500 < (*.f64 -2 x) < 2.0000000000000001e-4

   1. Initial program 58.5

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

   2. Taylor expanded in x around 0 0.5

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

   3. Applied egg-rr 0.5

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

Alternatives

Alternative 1
Error	16.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	46.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))