Logistic function from Lakshay Garg

Average Accuracy: 54.3% → 100.0%

Time: 13.4s

Precision: binary64

Cost: 27460

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{1}{1 + t_0} \cdot \left(-1 + {\left(\frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}\right)}^{2}\right)}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + -1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) -0.1)
     (/
      1.0
      (/
       1.0
       (*
        (/ 1.0 (+ 1.0 t_0))
        (+ -1.0 (pow (/ -2.0 (- -1.0 (pow (exp -2.0) x))) 2.0)))))
     (if (<= (* -2.0 x) 0.005)
       (+
        (* -0.05396825396825397 (pow x 7.0))
        (+
         (* -0.3333333333333333 (pow x 3.0))
         (+ x (* 0.13333333333333333 (pow x 5.0)))))
       (+ 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 = 2.0 / (1.0 + exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = 1.0 / (1.0 / ((1.0 / (1.0 + t_0)) * (-1.0 + pow((-2.0 / (-1.0 - pow(exp(-2.0), x))), 2.0))));
	} else if ((-2.0 * x) <= 0.005) {
		tmp = (-0.05396825396825397 * pow(x, 7.0)) + ((-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0))));
	} else {
		tmp = 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 = 2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))
    if (((-2.0d0) * x) <= (-0.1d0)) then
        tmp = 1.0d0 / (1.0d0 / ((1.0d0 / (1.0d0 + t_0)) * ((-1.0d0) + (((-2.0d0) / ((-1.0d0) - (exp((-2.0d0)) ** x))) ** 2.0d0))))
    else if (((-2.0d0) * x) <= 0.005d0) then
        tmp = ((-0.05396825396825397d0) * (x ** 7.0d0)) + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0))))
    else
        tmp = 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 = 2.0 / (1.0 + Math.exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -0.1) {
		tmp = 1.0 / (1.0 / ((1.0 / (1.0 + t_0)) * (-1.0 + Math.pow((-2.0 / (-1.0 - Math.pow(Math.exp(-2.0), x))), 2.0))));
	} else if ((-2.0 * x) <= 0.005) {
		tmp = (-0.05396825396825397 * Math.pow(x, 7.0)) + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0))));
	} else {
		tmp = 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 = 2.0 / (1.0 + math.exp((-2.0 * x)))
	tmp = 0
	if (-2.0 * x) <= -0.1:
		tmp = 1.0 / (1.0 / ((1.0 / (1.0 + t_0)) * (-1.0 + math.pow((-2.0 / (-1.0 - math.pow(math.exp(-2.0), x))), 2.0))))
	elif (-2.0 * x) <= 0.005:
		tmp = (-0.05396825396825397 * math.pow(x, 7.0)) + ((-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0))))
	else:
		tmp = 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 = Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.1)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(1.0 / Float64(1.0 + t_0)) * Float64(-1.0 + (Float64(-2.0 / Float64(-1.0 - (exp(-2.0) ^ x))) ^ 2.0)))));
	elseif (Float64(-2.0 * x) <= 0.005)
		tmp = Float64(Float64(-0.05396825396825397 * (x ^ 7.0)) + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0)))));
	else
		tmp = Float64(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 = 2.0 / (1.0 + exp((-2.0 * x)));
	tmp = 0.0;
	if ((-2.0 * x) <= -0.1)
		tmp = 1.0 / (1.0 / ((1.0 / (1.0 + t_0)) * (-1.0 + ((-2.0 / (-1.0 - (exp(-2.0) ^ x))) ^ 2.0))));
	elseif ((-2.0 * x) <= 0.005)
		tmp = (-0.05396825396825397 * (x ^ 7.0)) + ((-0.3333333333333333 * (x ^ 3.0)) + (x + (0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = 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[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.1], N[(1.0 / N[(1.0 / N[(N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[Power[N[(-2.0 / N[(-1.0 - N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005], N[(N[(-0.05396825396825397 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 100.0%

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

   if -0.10000000000000001 < (*.f64 -2 x) < 0.0050000000000000001

   1. Initial program 8.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 0.0050000000000000001 < (*.f64 -2 x)

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{1}{1 + \frac{2}{1 + e^{-2 \cdot x}}} \cdot \left(-1 + {\left(\frac{-2}{-1 - {\left(e^{-2}\right)}^{x}}\right)}^{2}\right)}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	21064

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ t_1 := t_0 + -1\\ \mathbf{if}\;t_0 \leq 0.999:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t_1}}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	20868

\[\begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := -1 - t_0\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{-1}{-1 - \frac{-2}{t_1}} \cdot \left(-1 + \frac{4}{{t_1}^{2}}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + t_0} + -1\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	20809

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;t_0 \leq 0.999 \lor \neg \left(t_0 \leq 1\right):\\ \;\;\;\;t_0 + -1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	20744

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -0.1:\\ \;\;\;\;\frac{1}{\frac{1}{t_0}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	100.0%
Cost	14024

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;\frac{1}{\frac{1}{t_0}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	79.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 7
Accuracy	78.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 8
Accuracy	55.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 9
Accuracy	55.4%
Cost	196

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

Alternative 10
Accuracy	7.0%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023141 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))