Logistic function from Lakshay Garg

Average Error: 28.9 → 0.9

Time: 7.4s

Precision: binary64

Cost: 20356

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;{\left(\frac{1}{\sqrt{\frac{-1}{1 + \frac{2}{-1 - e^{-2 \cdot x}}}}}\right)}^{2}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

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) -0.2)
   (pow (/ 1.0 (sqrt (/ -1.0 (+ 1.0 (/ 2.0 (- -1.0 (exp (* -2.0 x)))))))) 2.0)
   (if (<= (* -2.0 x) 1e-17) 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) <= -0.2) {
		tmp = pow((1.0 / sqrt((-1.0 / (1.0 + (2.0 / (-1.0 - exp((-2.0 * x)))))))), 2.0);
	} else if ((-2.0 * x) <= 1e-17) {
		tmp = 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) <= (-0.2d0)) then
        tmp = (1.0d0 / sqrt(((-1.0d0) / (1.0d0 + (2.0d0 / ((-1.0d0) - exp(((-2.0d0) * x)))))))) ** 2.0d0
    else if (((-2.0d0) * x) <= 1d-17) then
        tmp = 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) <= -0.2) {
		tmp = Math.pow((1.0 / Math.sqrt((-1.0 / (1.0 + (2.0 / (-1.0 - Math.exp((-2.0 * x)))))))), 2.0);
	} else if ((-2.0 * x) <= 1e-17) {
		tmp = 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) <= -0.2:
		tmp = math.pow((1.0 / math.sqrt((-1.0 / (1.0 + (2.0 / (-1.0 - math.exp((-2.0 * x)))))))), 2.0)
	elif (-2.0 * x) <= 1e-17:
		tmp = 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) <= -0.2)
		tmp = Float64(1.0 / sqrt(Float64(-1.0 / Float64(1.0 + Float64(2.0 / Float64(-1.0 - exp(Float64(-2.0 * x)))))))) ^ 2.0;
	elseif (Float64(-2.0 * x) <= 1e-17)
		tmp = 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) <= -0.2)
		tmp = (1.0 / sqrt((-1.0 / (1.0 + (2.0 / (-1.0 - exp((-2.0 * x)))))))) ^ 2.0;
	elseif ((-2.0 * x) <= 1e-17)
		tmp = 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], -0.2], N[Power[N[(1.0 / N[Sqrt[N[(-1.0 / N[(1.0 + N[(2.0 / N[(-1.0 - N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-17], x, -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.20000000000000001

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\frac{\frac{4}{{\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} + -1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}}} \]

   3. Applied egg-rr 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \frac{1}{\sqrt{\frac{-1}{1 - \frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}}}} \cdot \frac{1}{\sqrt{\frac{-1}{1 - \frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}}}} \]
      unpow2 [<=] 0.0	\[ \color{blue}{{\left(\frac{1}{\sqrt{\frac{-1}{1 - \frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}}}}\right)}^{2}} \]
      sub-neg [=>] 0.0	\[ {\left(\frac{1}{\sqrt{\frac{-1}{\color{blue}{1 + \left(-\frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}\right)}}}}\right)}^{2} \]
      distribute-neg-frac [=>] 0.0	\[ {\left(\frac{1}{\sqrt{\frac{-1}{1 + \color{blue}{\frac{--2}{-1 - {\left(e^{-2}\right)}^{x}}}}}}\right)}^{2} \]
      metadata-eval [=>] 0.0	\[ {\left(\frac{1}{\sqrt{\frac{-1}{1 + \frac{\color{blue}{2}}{-1 - {\left(e^{-2}\right)}^{x}}}}}\right)}^{2} \]
      exp-prod [<=] 0.0	\[ {\left(\frac{1}{\sqrt{\frac{-1}{1 + \frac{2}{-1 - \color{blue}{e^{-2 \cdot x}}}}}}\right)}^{2} \]
      *-commutative [=>] 0.0	\[ {\left(\frac{1}{\sqrt{\frac{-1}{1 + \frac{2}{-1 - e^{\color{blue}{x \cdot -2}}}}}}\right)}^{2} \]

   if -0.20000000000000001 < (*.f64 -2 x) < 1.00000000000000007e-17

   1. Initial program 59.7

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

   2. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{x} \]

   if 1.00000000000000007e-17 < (*.f64 -2 x)

   1. Initial program 1.4

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

   2. Applied egg-rr 1.4

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

   3. Taylor expanded in x around 0 3.5

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

   4. Simplified 3.5

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

      [Start] 3.5	\[ \frac{1}{\frac{1}{\frac{2}{1 + \left(1 + -2 \cdot x\right)} + -1}} \]
      *-commutative [=>] 3.5	\[ \frac{1}{\frac{1}{\frac{2}{1 + \left(1 + \color{blue}{x \cdot -2}\right)} + -1}} \]

   5. Taylor expanded in x around inf 3.1

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;{\left(\frac{1}{\sqrt{\frac{-1}{1 + \frac{2}{-1 - e^{-2 \cdot x}}}}}\right)}^{2}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	7492

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\frac{1}{\frac{1}{-1 + \frac{2}{1 + e^{-2 \cdot x}}}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7236

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	15.9
Cost	196

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

Alternative 4
Error	31.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))