Logistic function from Lakshay Garg

Average Error: 29.2 → 0.6

Time: 11.5s

Precision: binary64

Cost: 26180

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

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) -0.001)
   (expm1 (- (log 2.0) (log1p (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) <= -0.001) {
		tmp = expm1((log(2.0) - log1p(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) <= -0.001) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(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) <= -0.001:
		tmp = math.expm1((math.log(2.0) - math.log1p(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) <= -0.001)
		tmp = expm1(Float64(log(2.0) - log1p(exp(Float64(-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], -0.001], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

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

   1. Initial program 38.7

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

   2. Applied egg-rr 38.6

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	20164

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;{\left({\left(\frac{2}{e^{-2 \cdot x} + 1} + -1\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	7236

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.001:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3
Error	13.4
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 4
Error	13.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \]

Alternative 5
Error	13.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 6
Error	13.0
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	42.9
Cost	196

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

Alternative 8
Error	46.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))