Logistic function from Lakshay Garg

Average Error: 29.0 → 0.5

Time: 14.0s

Precision: binary64

Cost: 67144

Math

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

↓

\[\begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t_0}\\ t_2 := \sqrt[3]{1 + t_1}\\ \mathbf{if}\;-2 \cdot x \leq -100000000:\\ \;\;\;\;t_1 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4}{{t_0}^{2}} + -1}{{\left(\sqrt[3]{t_2 \cdot t_2}\right)}^{3}}}{t_2}\\ \end{array} \]

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 (cbrt (+ 1.0 t_1))))
   (if (<= (* -2.0 x) -100000000.0)
     (+ t_1 -1.0)
     (if (<= (* -2.0 x) 5e-8)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (/
        (/ (+ (/ 4.0 (pow t_0 2.0)) -1.0) (pow (cbrt (* t_2 t_2)) 3.0))
        t_2)))))

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 = cbrt((1.0 + t_1));
	double tmp;
	if ((-2.0 * x) <= -100000000.0) {
		tmp = t_1 + -1.0;
	} else if ((-2.0 * x) <= 5e-8) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = (((4.0 / pow(t_0, 2.0)) + -1.0) / pow(cbrt((t_2 * t_2)), 3.0)) / t_2;
	}
	return tmp;
}

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 = Math.cbrt((1.0 + t_1));
	double tmp;
	if ((-2.0 * x) <= -100000000.0) {
		tmp = t_1 + -1.0;
	} else if ((-2.0 * x) <= 5e-8) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} else {
		tmp = (((4.0 / Math.pow(t_0, 2.0)) + -1.0) / Math.pow(Math.cbrt((t_2 * t_2)), 3.0)) / t_2;
	}
	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 = cbrt(Float64(1.0 + t_1))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -100000000.0)
		tmp = Float64(t_1 + -1.0);
	elseif (Float64(-2.0 * x) <= 5e-8)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = Float64(Float64(Float64(Float64(4.0 / (t_0 ^ 2.0)) + -1.0) / (cbrt(Float64(t_2 * t_2)) ^ 3.0)) / t_2);
	end
	return 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[Power[N[(1.0 + t$95$1), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -100000000.0], N[(t$95$1 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 5e-8], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(4.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Power[N[Power[N[(t$95$2 * t$95$2), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -1e8

   1. Initial program 0

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

   if -1e8 < (*.f64 -2 x) < 4.9999999999999998e-8

   1. Initial program 58.6

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

   2. Taylor expanded in x around 0 0.8

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + x} \]

   if 4.9999999999999998e-8 < (*.f64 -2 x)

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} + -1}{\sqrt[3]{1 + \frac{2}{1 + e^{-2 \cdot x}}} \cdot \sqrt[3]{1 + \frac{2}{1 + e^{-2 \cdot x}}}}}{\sqrt[3]{1 + \frac{2}{1 + e^{-2 \cdot x}}}}} \]

   3. Applied egg-rr 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4}{{\left(1 + e^{-2 \cdot x}\right)}^{2}} + -1}{{\left(\sqrt[3]{\sqrt[3]{1 + \frac{2}{1 + e^{-2 \cdot x}}} \cdot \sqrt[3]{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\right)}^{3}}}{\sqrt[3]{1 + \frac{2}{1 + e^{-2 \cdot x}}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	47304

\[\begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \frac{2}{t_0}\\ t_2 := \sqrt[3]{1 + t_1}\\ \mathbf{if}\;-2 \cdot x \leq -100000000:\\ \;\;\;\;t_1 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{4}{{t_0}^{2}} + -1}{{t_2}^{2}}}{t_2}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	7624

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -100000000:\\ \;\;\;\;t_0 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-2 + \left(1 + t_0\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	7497

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -100000000 \lor \neg \left(-2 \cdot x \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 4
Error	13.9
Cost	708

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

Alternative 5
Error	13.5
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 6
Error	13.5
Cost	328

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

Alternative 7
Error	43.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 8
Error	46.5
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023011 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))