Logistic function from Lakshay Garg

Average Error: 29.3 → 0.3

Time: 8.5s

Precision: binary64

Cost: 7304

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

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) -0.1)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (if (<= (* -2.0 x) 4e-5) (+ (* -0.3333333333333333 (pow x 3.0)) 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) <= -0.1) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if ((-2.0 * x) <= 4e-5) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + 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) <= (-0.1d0)) then
        tmp = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
    else if (((-2.0d0) * x) <= 4d-5) then
        tmp = ((-0.3333333333333333d0) * (x ** 3.0d0)) + 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) <= -0.1) {
		tmp = (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
	} else if ((-2.0 * x) <= 4e-5) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + 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) <= -0.1:
		tmp = (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0
	elif (-2.0 * x) <= 4e-5:
		tmp = (-0.3333333333333333 * math.pow(x, 3.0)) + 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) <= -0.1)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (Float64(-2.0 * x) <= 4e-5)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + 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) <= -0.1)
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	elseif ((-2.0 * x) <= 4e-5)
		tmp = (-0.3333333333333333 * (x ^ 3.0)) + 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], -0.1], 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], 4e-5], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.10000000000000001

   1. Initial program 0.0

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

   if -0.10000000000000001 < (*.f64 -2 x) < 4.00000000000000033e-5

   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 4.00000000000000033e-5 < (*.f64 -2 x)

   1. Initial program 0.1

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

   2. Taylor expanded in x around 0 2.3

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

   3. Simplified 2.3

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

      [Start] 2.3	\[ \frac{2}{2 + -2 \cdot x} - 1 \]
      rational.json-simplify-2 [=>] 2.3	\[ \frac{2}{2 + \color{blue}{x \cdot -2}} - 1 \]

   4. Taylor expanded in x around inf 1.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	15.2
Cost	196

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

Alternative 2
Error	46.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))