Logistic function from Lakshay Garg

Average Error: 29.5 → 0.6

Time: 5.9s

Precision: binary64

Cost: 32516

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

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) -1e-6)
   (expm1 (- (log 2.0) (log1p (pow (exp -2.0) x))))
   (expm1 x)))

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) <= -1e-6) {
		tmp = expm1((log(2.0) - log1p(pow(exp(-2.0), x))));
	} else {
		tmp = expm1(x);
	}
	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 tmp;
	if ((-2.0 * x) <= -1e-6) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.pow(Math.exp(-2.0), x))));
	} else {
		tmp = Math.expm1(x);
	}
	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) <= -1e-6:
		tmp = math.expm1((math.log(2.0) - math.log1p(math.pow(math.exp(-2.0), x))))
	else:
		tmp = math.expm1(x)
	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) <= -1e-6)
		tmp = expm1(Float64(log(2.0) - log1p((exp(-2.0) ^ x))));
	else
		tmp = expm1(x);
	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_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -1e-6], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -9.99999999999999955e-7

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

   if -9.99999999999999955e-7 < (*.f64 -2 x)

   1. Initial program 39.5

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

   2. Applied egg-rr 39.4

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

   3. Taylor expanded in x around 0 0.8

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	7364

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;-2 + \left(1 + \frac{2}{2 + \mathsf{expm1}\left(-2 \cdot x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	7236

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3
Error	15.5
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -0.73:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	15.5
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 2022343 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))