Logistic function from Lakshay Garg

Average Error: 29.7 → 1.0

Time: 3.7s

Precision: binary64

Cost: 7816

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(-0.3333333333333333 \cdot x\right) + \left(0.13333333333333333 \cdot {x}^{5} + x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -1000000.0)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (if (<= (* -2.0 x) 2e-15)
     (+
      (* (* x x) (* -0.3333333333333333 x))
      (+ (* 0.13333333333333333 (pow x 5.0)) x))
     -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 tmp;
	if ((-2.0 * x) <= -1000000.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if ((-2.0 * x) <= 2e-15) {
		tmp = ((x * x) * (-0.3333333333333333 * x)) + ((0.13333333333333333 * pow(x, 5.0)) + x);
	} 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) :: tmp
    if (((-2.0d0) * x) <= (-1000000.0d0)) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
    else if (((-2.0d0) * x) <= 2d-15) then
        tmp = ((x * x) * ((-0.3333333333333333d0) * x)) + ((0.13333333333333333d0 * (x ** 5.0d0)) + x)
    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 tmp;
	if ((-2.0 * x) <= -1000000.0) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
	} else if ((-2.0 * x) <= 2e-15) {
		tmp = ((x * x) * (-0.3333333333333333 * x)) + ((0.13333333333333333 * Math.pow(x, 5.0)) + x);
	} 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):
	tmp = 0
	if (-2.0 * x) <= -1000000.0:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0
	elif (-2.0 * x) <= 2e-15:
		tmp = ((x * x) * (-0.3333333333333333 * x)) + ((0.13333333333333333 * math.pow(x, 5.0)) + x)
	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)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1000000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (Float64(-2.0 * x) <= 2e-15)
		tmp = Float64(Float64(Float64(x * x) * Float64(-0.3333333333333333 * x)) + Float64(Float64(0.13333333333333333 * (x ^ 5.0)) + x));
	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)
	tmp = 0.0;
	if ((-2.0 * x) <= -1000000.0)
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	elseif ((-2.0 * x) <= 2e-15)
		tmp = ((x * x) * (-0.3333333333333333 * x)) + ((0.13333333333333333 * (x ^ 5.0)) + x);
	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_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1000000.0], 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], 2e-15], N[(N[(N[(x * x), $MachinePrecision] * N[(-0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -1e6

   1. Initial program 0

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

   if -1e6 < (*.f64 -2 x) < 2.0000000000000002e-15

   1. Initial program 58.8

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

   2. Taylor expanded in x around 0 0.6

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

   3. Applied egg-rr 0.6

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

   if 2.0000000000000002e-15 < (*.f64 -2 x)

   1. Initial program 1.0

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

   2. Taylor expanded in x around 0 3.1

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

   3. Taylor expanded in x around inf 2.7

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

Alternatives

Alternative 1
Error	0.7
Cost	7236

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.005:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(-0.3333333333333333 \cdot \left(x \cdot x\right)\right) + x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	15.6
Cost	324

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	30.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))