Logistic function from Lakshay Garg

Average Error: 28.1 → 0.1

Time: 12.2s

Precision: binary64

Cost: 13764

Math

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

↓

\[\begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{t_0 + -1}{-1 - t_0}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + t_0} - 1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))))
   (if (<= (* -2.0 x) -0.05)
     (/ (+ t_0 -1.0) (- -1.0 t_0))
     (if (<= (* -2.0 x) 2e-8)
       (+ (* -0.3333333333333333 (pow x 3.0)) x)
       (- (/ 2.0 (+ 1.0 t_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 = exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (t_0 + -1.0) / (-1.0 - t_0);
	} else if ((-2.0 * x) <= 2e-8) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + x;
	} else {
		tmp = (2.0 / (1.0 + t_0)) - 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 = exp(((-2.0d0) * x))
    if (((-2.0d0) * x) <= (-0.05d0)) then
        tmp = (t_0 + (-1.0d0)) / ((-1.0d0) - t_0)
    else if (((-2.0d0) * x) <= 2d-8) then
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + x
    else
        tmp = (2.0d0 / (1.0d0 + t_0)) - 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 = Math.exp((-2.0 * x));
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (t_0 + -1.0) / (-1.0 - t_0);
	} else if ((-2.0 * x) <= 2e-8) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + x;
	} else {
		tmp = (2.0 / (1.0 + t_0)) - 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 = math.exp((-2.0 * x))
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = (t_0 + -1.0) / (-1.0 - t_0)
	elif (-2.0 * x) <= 2e-8:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + x
	else:
		tmp = (2.0 / (1.0 + t_0)) - 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 = exp(Float64(-2.0 * x))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(t_0 + -1.0) / Float64(-1.0 - t_0));
	elseif (Float64(-2.0 * x) <= 2e-8)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + x);
	else
		tmp = Float64(Float64(2.0 / Float64(1.0 + t_0)) - 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 = exp((-2.0 * x));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.05)
		tmp = (t_0 + -1.0) / (-1.0 - t_0);
	elseif ((-2.0 * x) <= 2e-8)
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + x;
	else
		tmp = (2.0 / (1.0 + t_0)) - 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[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2e-8], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Applied egg-rr 0.0

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

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

   1. Initial program 59.3

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

   2. Taylor expanded in x around 0 0.0

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

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

   1. Initial program 0.3

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{e^{-2 \cdot x} + -1}{-1 - e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7496

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{1}{-1 + \frac{-2}{-1 + e^{x \cdot -2}}}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array} \]

Alternative 3
Error	13.9
Cost	7304

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20000:\\ \;\;\;\;\frac{e^{-2 \cdot x} + -1}{-2}\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-8}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	15.8
Cost	1220

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(1 + \left(\frac{1}{x} + \left(\left(x \cdot 0.3333333333333333 + 1\right) - 1\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	15.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x} + 0.3333333333333333 \cdot x}\\ \end{array} \]

Alternative 6
Error	16.0
Cost	196

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

Alternative 7
Error	45.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023067 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))