Logistic function from Lakshay Garg

Average Error: 29.0 → 0.5

Time: 1.9s

Precision: binary64

Cost: 14024

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-15}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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) -100000.0)
     t_0
     (if (<= (* -2.0 x) 5e-15)
       (+
        (* -0.3333333333333333 (pow x 3.0))
        (+ (* 0.13333333333333333 (pow x 5.0)) 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) <= -100000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 5e-15) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + ((0.13333333333333333 * pow(x, 5.0)) + 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) <= (-100000.0d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 5d-15) then
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + ((0.13333333333333333d0 * (x ** 5.0d0)) + 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) <= -100000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 5e-15) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + ((0.13333333333333333 * Math.pow(x, 5.0)) + 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) <= -100000.0:
		tmp = t_0
	elif (-2.0 * x) <= 5e-15:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + ((0.13333333333333333 * math.pow(x, 5.0)) + 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) <= -100000.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 5e-15)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(Float64(0.13333333333333333 * (x ^ 5.0)) + 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) <= -100000.0)
		tmp = t_0;
	elseif ((-2.0 * x) <= 5e-15)
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + ((0.13333333333333333 * (x ^ 5.0)) + 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], -100000.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-15], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1e5 or 4.99999999999999999e-15 < (*.f64 -2 x)

   1. Initial program 0.4

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

   if -1e5 < (*.f64 -2 x) < 4.99999999999999999e-15

   1. Initial program 58.9

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

   2. Taylor expanded in x around 0 0.5

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-15}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	20808

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ t_1 := t_0 - 1\\ \mathbf{if}\;t_0 \leq 0.95:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 1.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.0
Cost	196

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

Alternative 3
Error	46.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023053 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))