Logistic function from Lakshay Garg

Average Error: 29.2 → 0.4

Time: 6.9s

Precision: binary64

Cost: 20744

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;0.13333333333333333 \cdot {x}^{5} + \left(x + \left(-0.05396825396825397 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (/ x -0.5)))) -1.0)))
   (if (<= (* -2.0 x) -200.0)
     t_0
     (if (<= (* -2.0 x) 5e-18)
       (+
        (* 0.13333333333333333 (pow x 5.0))
        (+
         x
         (+
          (* -0.05396825396825397 (pow x 7.0))
          (* -0.3333333333333333 (pow x 3.0)))))
       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((x / -0.5)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -200.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 5e-18) {
		tmp = (0.13333333333333333 * pow(x, 5.0)) + (x + ((-0.05396825396825397 * pow(x, 7.0)) + (-0.3333333333333333 * pow(x, 3.0))));
	} 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((x / (-0.5d0))))) + (-1.0d0)
    if (((-2.0d0) * x) <= (-200.0d0)) then
        tmp = t_0
    else if (((-2.0d0) * x) <= 5d-18) then
        tmp = (0.13333333333333333d0 * (x ** 5.0d0)) + (x + (((-0.05396825396825397d0) * (x ** 7.0d0)) + ((-0.3333333333333333d0) * (x ** 3.0d0))))
    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((x / -0.5)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -200.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 5e-18) {
		tmp = (0.13333333333333333 * Math.pow(x, 5.0)) + (x + ((-0.05396825396825397 * Math.pow(x, 7.0)) + (-0.3333333333333333 * Math.pow(x, 3.0))));
	} 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((x / -0.5)))) + -1.0
	tmp = 0
	if (-2.0 * x) <= -200.0:
		tmp = t_0
	elif (-2.0 * x) <= 5e-18:
		tmp = (0.13333333333333333 * math.pow(x, 5.0)) + (x + ((-0.05396825396825397 * math.pow(x, 7.0)) + (-0.3333333333333333 * math.pow(x, 3.0))))
	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(x / -0.5)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -200.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 5e-18)
		tmp = Float64(Float64(0.13333333333333333 * (x ^ 5.0)) + Float64(x + Float64(Float64(-0.05396825396825397 * (x ^ 7.0)) + Float64(-0.3333333333333333 * (x ^ 3.0)))));
	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((x / -0.5)))) + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -200.0)
		tmp = t_0;
	elseif ((-2.0 * x) <= 5e-18)
		tmp = (0.13333333333333333 * (x ^ 5.0)) + (x + ((-0.05396825396825397 * (x ^ 7.0)) + (-0.3333333333333333 * (x ^ 3.0))));
	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[(x / -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -200.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-18], N[(N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(-0.05396825396825397 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -200 or 5.00000000000000036e-18 < (*.f64 -2 x)

   1. Initial program 0.7

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1} \]

   if -200 < (*.f64 -2 x) < 5.00000000000000036e-18

   1. Initial program 59.2

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

   2. Simplified 59.2

      \[\leadsto \color{blue}{\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1} \]

   3. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)\right)} \]

   4. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(0.13333333333333333, {x}^{5}, x\right), \mathsf{fma}\left(-0.05396825396825397, {x}^{7}, -0.3333333333333333 \cdot {x}^{3}\right)\right)} \]

   5. Applied egg-rr 0.2

      \[\leadsto \color{blue}{0.13333333333333333 \cdot {x}^{5} + \left(x + \mathsf{fma}\left(-0.3333333333333333, {x}^{3}, -0.05396825396825397 \cdot {x}^{7}\right)\right)} \]

   6. Taylor expanded in x around 0 0.2

      \[\leadsto 0.13333333333333333 \cdot {x}^{5} + \left(x + \color{blue}{\left(-0.05396825396825397 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;0.13333333333333333 \cdot {x}^{5} + \left(x + \left(-0.05396825396825397 \cdot {x}^{7} + -0.3333333333333333 \cdot {x}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	14024

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.4
Cost	7496

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{\frac{x}{-0.5}}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -0.005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	14.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	13.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.46340488633879:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.005317090888249833:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 5
Error	13.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -8.46340488633879:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.005317090888249833:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 6
Error	28.6
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 0.005317090888249833:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Error	59.5
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022224 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))