Logistic function from Lakshay Garg

Average Error: 29.5 → 0.3

Time: 8.7s

Precision: binary64

Cost: 32516

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

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) -0.02)
   (expm1 (- (log 2.0) (log1p (pow (exp -2.0) x))))
   (if (<= (* -2.0 x) 4e-7)
     (+
      (* -0.05396825396825397 (pow x 7.0))
      (+
       (* -0.3333333333333333 (pow x 3.0))
       (+ x (* 0.13333333333333333 (pow x 5.0)))))
     -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) <= -0.02) {
		tmp = expm1((log(2.0) - log1p(pow(exp(-2.0), x))));
	} else if ((-2.0 * x) <= 4e-7) {
		tmp = (-0.05396825396825397 * pow(x, 7.0)) + ((-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0))));
	} 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) <= -0.02) {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.pow(Math.exp(-2.0), x))));
	} else if ((-2.0 * x) <= 4e-7) {
		tmp = (-0.05396825396825397 * Math.pow(x, 7.0)) + ((-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0))));
	} 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) <= -0.02:
		tmp = math.expm1((math.log(2.0) - math.log1p(math.pow(math.exp(-2.0), x))))
	elif (-2.0 * x) <= 4e-7:
		tmp = (-0.05396825396825397 * math.pow(x, 7.0)) + ((-0.3333333333333333 * math.pow(x, 3.0)) + (x + (0.13333333333333333 * math.pow(x, 5.0))))
	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) <= -0.02)
		tmp = expm1(Float64(log(2.0) - log1p((exp(-2.0) ^ x))));
	elseif (Float64(-2.0 * x) <= 4e-7)
		tmp = Float64(Float64(-0.05396825396825397 * (x ^ 7.0)) + Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0)))));
	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], -0.02], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Power[N[Exp[-2.0], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 4e-7], N[(N[(-0.05396825396825397 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -0.0200000000000000004

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   if -0.0200000000000000004 < (*.f64 -2 x) < 3.9999999999999998e-7

   1. Initial program 59.3

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

   2. Taylor expanded in x around 0 0

      \[\leadsto \color{blue}{-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)\right)} \]

   if 3.9999999999999998e-7 < (*.f64 -2 x)

   1. Initial program 0.2

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

   2. Taylor expanded in x around 0 2.3

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

   3. Simplified 2.3

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

      [Start] 2.3	\[ \frac{2}{2 + -2 \cdot x} - 1 \]
      *-commutative [=>] 2.3	\[ \frac{2}{2 + \color{blue}{x \cdot -2}} - 1 \]

   4. Taylor expanded in x around inf 1.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	20744

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2000000000:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	0.3
Cost	14024

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.02:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7304

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.005:\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{elif}\;-2 \cdot x \leq 4 \cdot 10^{-7}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	13.7
Cost	708

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

Alternative 5
Error	13.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 2.55:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 - \frac{4}{x}\\ \end{array} \]

Alternative 6
Error	13.4
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	43.6
Cost	196

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

Alternative 8
Error	46.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))