Logistic function from Lakshay Garg

Average Error: 45.39% → 0.24%

Time: 11.2s

Precision: binary64

Cost: 27780

Math

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

↓

\[\begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := 1 + \frac{2}{t_0}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{4}{t_1 \cdot {t_0}^{2}} + \frac{-1}{t_1}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))) (t_1 (+ 1.0 (/ 2.0 t_0))))
   (if (<= (* -2.0 x) -0.05)
     (+ (/ 4.0 (* t_1 (pow t_0 2.0))) (/ -1.0 t_1))
     (if (<= (* -2.0 x) 0.01)
       (+
        (* -0.05396825396825397 (pow x 7.0))
        (+
         (* -0.3333333333333333 (pow x 3.0))
         (+ x (* 0.13333333333333333 (pow x 5.0)))))
       -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 = 1.0 + exp((-2.0 * x));
	double t_1 = 1.0 + (2.0 / t_0);
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (4.0 / (t_1 * pow(t_0, 2.0))) + (-1.0 / t_1);
	} else if ((-2.0 * x) <= 0.01) {
		tmp = (-0.05396825396825397 * pow(x, 7.0)) + ((-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0))));
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + exp(((-2.0d0) * x))
    t_1 = 1.0d0 + (2.0d0 / t_0)
    if (((-2.0d0) * x) <= (-0.05d0)) then
        tmp = (4.0d0 / (t_1 * (t_0 ** 2.0d0))) + ((-1.0d0) / t_1)
    else if (((-2.0d0) * x) <= 0.01d0) then
        tmp = ((-0.05396825396825397d0) * (x ** 7.0d0)) + (((-0.3333333333333333d0) * (x ** 3.0d0)) + (x + (0.13333333333333333d0 * (x ** 5.0d0))))
    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 t_0 = 1.0 + Math.exp((-2.0 * x));
	double t_1 = 1.0 + (2.0 / t_0);
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = (4.0 / (t_1 * Math.pow(t_0, 2.0))) + (-1.0 / t_1);
	} else if ((-2.0 * x) <= 0.01) {
		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 = -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 = 1.0 + math.exp((-2.0 * x))
	t_1 = 1.0 + (2.0 / t_0)
	tmp = 0
	if (-2.0 * x) <= -0.05:
		tmp = (4.0 / (t_1 * math.pow(t_0, 2.0))) + (-1.0 / t_1)
	elif (-2.0 * x) <= 0.01:
		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 = -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(1.0 + exp(Float64(-2.0 * x)))
	t_1 = Float64(1.0 + Float64(2.0 / t_0))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = Float64(Float64(4.0 / Float64(t_1 * (t_0 ^ 2.0))) + Float64(-1.0 / t_1));
	elseif (Float64(-2.0 * x) <= 0.01)
		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 = -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 = 1.0 + exp((-2.0 * x));
	t_1 = 1.0 + (2.0 / t_0);
	tmp = 0.0;
	if ((-2.0 * x) <= -0.05)
		tmp = (4.0 / (t_1 * (t_0 ^ 2.0))) + (-1.0 / t_1);
	elseif ((-2.0 * x) <= 0.01)
		tmp = (-0.05396825396825397 * (x ^ 7.0)) + ((-0.3333333333333333 * (x ^ 3.0)) + (x + (0.13333333333333333 * (x ^ 5.0))));
	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_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[(N[(4.0 / N[(t$95$1 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], 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], -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.050000000000000003

   1. Initial program 0.01

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

   2. Applied egg-rr 0.01

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

   3. Applied egg-rr 0.01

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

   4. Simplified 0.01

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

      [Start] 0.01	\[ \frac{4}{\left(1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}\right) \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} - \frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]
      exp-prod [<=] 0.01	\[ \frac{4}{\left(1 + \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}}\right) \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} - \frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]
      *-commutative [=>] 0.01	\[ \frac{4}{\left(1 + \frac{2}{1 + e^{\color{blue}{x \cdot -2}}}\right) \cdot {\left(1 + {\left(e^{-2}\right)}^{x}\right)}^{2}} - \frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]
      exp-prod [<=] 0.01	\[ \frac{4}{\left(1 + \frac{2}{1 + e^{x \cdot -2}}\right) \cdot {\left(1 + \color{blue}{e^{-2 \cdot x}}\right)}^{2}} - \frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]
      *-commutative [=>] 0.01	\[ \frac{4}{\left(1 + \frac{2}{1 + e^{x \cdot -2}}\right) \cdot {\left(1 + e^{\color{blue}{x \cdot -2}}\right)}^{2}} - \frac{1}{1 + \frac{2}{1 + {\left(e^{-2}\right)}^{x}}} \]
      exp-prod [<=] 0.01	\[ \frac{4}{\left(1 + \frac{2}{1 + e^{x \cdot -2}}\right) \cdot {\left(1 + e^{x \cdot -2}\right)}^{2}} - \frac{1}{1 + \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}}} \]
      *-commutative [=>] 0.01	\[ \frac{4}{\left(1 + \frac{2}{1 + e^{x \cdot -2}}\right) \cdot {\left(1 + e^{x \cdot -2}\right)}^{2}} - \frac{1}{1 + \frac{2}{1 + e^{\color{blue}{x \cdot -2}}}} \]

   if -0.050000000000000003 < (*.f64 -2 x) < 0.0100000000000000002

   1. Initial program 92.21

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

   2. Taylor expanded in x around 0 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 0.0100000000000000002 < (*.f64 -2 x)

   1. Initial program 0.03

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

   2. Taylor expanded in x around 0 2.44

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

   3. Simplified 2.44

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

      [Start] 2.44	\[ \frac{2}{2 + -2 \cdot x} - 1 \]
      *-commutative [=>] 2.44	\[ \frac{2}{2 + \color{blue}{x \cdot -2}} - 1 \]

   4. Taylor expanded in x around inf 0.93

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

4. Final simplification 0.24

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{4}{\left(1 + \frac{2}{1 + e^{-2 \cdot x}}\right) \cdot {\left(1 + e^{-2 \cdot x}\right)}^{2}} + \frac{-1}{1 + \frac{2}{1 + e^{-2 \cdot x}}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.24%
Cost	20744

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	0.25%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} + -1}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	0.24%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	0.14%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05 \lor \neg \left(-2 \cdot x \leq 10^{-7}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 5
Error	24.53%
Cost	196

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

Alternative 6
Error	72.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))