Logistic function from Lakshay Garg

Average Error: 29.8 → 0.1

Time: 13.9s

Precision: binary64

Cost: 35336

Math

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

↓

\[\begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t_0}\\ t_2 := t_1 + -1\\ \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;t_1 - 1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} + -0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_2}}{t_2 \cdot t_2} \cdot {\left(2 \cdot \frac{1}{t_0} - 1\right)}^{4}\\ \end{array} \]

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 (/ 2.0 t_0)) (t_2 (+ t_1 -1.0)))
   (if (<= (* -2.0 x) -1.0)
     (- t_1 1.0)
     (if (<= (* -2.0 x) 0.002)
       (+
        x
        (+
         (* 0.13333333333333333 (pow x 5.0))
         (* -0.3333333333333333 (pow x 3.0))))
       (*
        (/ (/ 1.0 t_2) (* t_2 t_2))
        (pow (- (* 2.0 (/ 1.0 t_0)) 1.0) 4.0))))))

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 = 2.0 / t_0;
	double t_2 = t_1 + -1.0;
	double tmp;
	if ((-2.0 * x) <= -1.0) {
		tmp = t_1 - 1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x + ((0.13333333333333333 * pow(x, 5.0)) + (-0.3333333333333333 * pow(x, 3.0)));
	} else {
		tmp = ((1.0 / t_2) / (t_2 * t_2)) * pow(((2.0 * (1.0 / t_0)) - 1.0), 4.0);
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + exp(((-2.0d0) * x))
    t_1 = 2.0d0 / t_0
    t_2 = t_1 + (-1.0d0)
    if (((-2.0d0) * x) <= (-1.0d0)) then
        tmp = t_1 - 1.0d0
    else if (((-2.0d0) * x) <= 0.002d0) then
        tmp = x + ((0.13333333333333333d0 * (x ** 5.0d0)) + ((-0.3333333333333333d0) * (x ** 3.0d0)))
    else
        tmp = ((1.0d0 / t_2) / (t_2 * t_2)) * (((2.0d0 * (1.0d0 / t_0)) - 1.0d0) ** 4.0d0)
    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 = 2.0 / t_0;
	double t_2 = t_1 + -1.0;
	double tmp;
	if ((-2.0 * x) <= -1.0) {
		tmp = t_1 - 1.0;
	} else if ((-2.0 * x) <= 0.002) {
		tmp = x + ((0.13333333333333333 * Math.pow(x, 5.0)) + (-0.3333333333333333 * Math.pow(x, 3.0)));
	} else {
		tmp = ((1.0 / t_2) / (t_2 * t_2)) * Math.pow(((2.0 * (1.0 / t_0)) - 1.0), 4.0);
	}
	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 = 2.0 / t_0
	t_2 = t_1 + -1.0
	tmp = 0
	if (-2.0 * x) <= -1.0:
		tmp = t_1 - 1.0
	elif (-2.0 * x) <= 0.002:
		tmp = x + ((0.13333333333333333 * math.pow(x, 5.0)) + (-0.3333333333333333 * math.pow(x, 3.0)))
	else:
		tmp = ((1.0 / t_2) / (t_2 * t_2)) * math.pow(((2.0 * (1.0 / t_0)) - 1.0), 4.0)
	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(2.0 / t_0)
	t_2 = Float64(t_1 + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1.0)
		tmp = Float64(t_1 - 1.0);
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(x + Float64(Float64(0.13333333333333333 * (x ^ 5.0)) + Float64(-0.3333333333333333 * (x ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / t_2) / Float64(t_2 * t_2)) * (Float64(Float64(2.0 * Float64(1.0 / t_0)) - 1.0) ^ 4.0));
	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 = 2.0 / t_0;
	t_2 = t_1 + -1.0;
	tmp = 0.0;
	if ((-2.0 * x) <= -1.0)
		tmp = t_1 - 1.0;
	elseif ((-2.0 * x) <= 0.002)
		tmp = x + ((0.13333333333333333 * (x ^ 5.0)) + (-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = ((1.0 / t_2) / (t_2 * t_2)) * (((2.0 * (1.0 / t_0)) - 1.0) ^ 4.0);
	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[(2.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1.0], N[(t$95$1 - 1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(x + N[(N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / t$95$2), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.0

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

   if -1 < (*.f64 -2 x) < 2e-3

   1. Initial program 58.9

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

   2. Taylor expanded in x around 0 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ -0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right) \]
      rational_best-simplify-43 [=>] 0.1	\[ \color{blue}{x + \left(0.13333333333333333 \cdot {x}^{5} + -0.3333333333333333 \cdot {x}^{3}\right)} \]

   if 2e-3 < (*.f64 -2 x)

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Taylor expanded in x around inf 0.0

      \[\leadsto \frac{\frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + -1}}{\left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right)} \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right) \cdot \color{blue}{{\left(2 \cdot \frac{1}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\right) \]

   4. Taylor expanded in x around inf 0.0

      \[\leadsto \frac{\frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + -1}}{\left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right)} \cdot \color{blue}{{\left(2 \cdot \frac{1}{1 + e^{-2 \cdot x}} - 1\right)}^{4}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} + -0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{1 + e^{-2 \cdot x}} + -1}}{\left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + -1\right)} \cdot {\left(2 \cdot \frac{1}{1 + e^{-2 \cdot x}} - 1\right)}^{4}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	21704

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ t_1 := t_0 + -1\\ \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;t_0 - 1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} + -0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_1 \cdot \frac{1}{t_1}\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	20808

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ t_1 := t_0 - 1\\ \mathbf{if}\;t_0 \leq 0.9995:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 1.5:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.1
Cost	14024

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;-2 \cdot x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;x + \left(0.13333333333333333 \cdot {x}^{5} + -0.3333333333333333 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	15.3
Cost	196

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

Alternative 5
Error	46.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023092 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))