Logistic function from Lakshay Garg

Average Error: 29.6 → 0.7

Time: 9.9s

Precision: binary64

Cost: 26436

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\mathsf{expm1}\left(\frac{1}{\frac{1}{\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(e^{-2 \cdot x}\right)}}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.2)
   (expm1 (/ 1.0 (/ 1.0 (- (log1p 1.0) (log1p (exp (* -2.0 x)))))))
   (if (<= (* -2.0 x) 1e-12) (+ x (* -0.3333333333333333 (pow x 3.0))) -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 tmp;
	if ((-2.0 * x) <= -0.2) {
		tmp = expm1((1.0 / (1.0 / (log1p(1.0) - log1p(exp((-2.0 * x)))))));
	} else if ((-2.0 * x) <= 1e-12) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = -1.0;
	}
	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) <= -0.2) {
		tmp = Math.expm1((1.0 / (1.0 / (Math.log1p(1.0) - Math.log1p(Math.exp((-2.0 * x)))))));
	} else if ((-2.0 * x) <= 1e-12) {
		tmp = x + (-0.3333333333333333 * Math.pow(x, 3.0));
	} 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):
	tmp = 0
	if (-2.0 * x) <= -0.2:
		tmp = math.expm1((1.0 / (1.0 / (math.log1p(1.0) - math.log1p(math.exp((-2.0 * x)))))))
	elif (-2.0 * x) <= 1e-12:
		tmp = x + (-0.3333333333333333 * math.pow(x, 3.0))
	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)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.2)
		tmp = expm1(Float64(1.0 / Float64(1.0 / Float64(log1p(1.0) - log1p(exp(Float64(-2.0 * x)))))));
	elseif (Float64(-2.0 * x) <= 1e-12)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = -1.0;
	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], -0.2], N[(Exp[N[(1.0 / N[(1.0 / N[(N[Log[1 + 1.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 1e-12], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.20000000000000001

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Applied egg-rr 0.0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{1}{\frac{1}{\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)}}}\right) \]

   4. Taylor expanded in x around inf 0.0

      \[\leadsto \mathsf{expm1}\left(\frac{1}{\frac{1}{\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(\color{blue}{e^{-2 \cdot x}}\right)}}\right) \]

   if -0.20000000000000001 < (*.f64 -2 x) < 9.9999999999999998e-13

   1. Initial program 59.5

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

   2. Taylor expanded in x around 0 0.1

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

   if 9.9999999999999998e-13 < (*.f64 -2 x)

   1. Initial program 0.8

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

   2. Taylor expanded in x around 0 2.6

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

   3. Simplified 2.6

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

      [Start] 2.6	\[ \frac{2}{2 + -2 \cdot x} - 1 \]
      *-commutative [=>] 2.6	\[ \frac{2}{2 + \color{blue}{x \cdot -2}} - 1 \]

   4. Taylor expanded in x around inf 2.3

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\mathsf{expm1}\left(\frac{1}{\frac{1}{\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(e^{-2 \cdot x}\right)}}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	20036

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{\mathsf{expm1}\left(x \cdot -4\right)}, \mathsf{expm1}\left(-2 \cdot x\right), -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	0.7
Cost	13892

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;-1 + \frac{2}{\frac{-1 + e^{x \cdot -4}}{\mathsf{expm1}\left(-2 \cdot x\right)}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7304

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-12}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	13.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\frac{1}{0.5 + \frac{1}{x}}\right)\\ \end{array} \]

Alternative 5
Error	15.1
Cost	196

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

Alternative 6
Error	46.4
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023045 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))