Logistic function from Lakshay Garg

Average Error: 29.5 → 0.4

Time: 7.0s

Precision: binary64

Cost: 26696

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, 0.13333333333333333 \cdot {x}^{5}\right), 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

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) -200000.0)
   (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)
   (if (<= (* -2.0 x) 0.005)
     (fma
      (fma -0.3333333333333333 (pow x 3.0) (* 0.13333333333333333 (pow x 5.0)))
      1.0
      x)
     -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) <= -200000.0) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	} else if ((-2.0 * x) <= 0.005) {
		tmp = fma(fma(-0.3333333333333333, pow(x, 3.0), (0.13333333333333333 * pow(x, 5.0))), 1.0, x);
	} 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) <= -200000.0)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0);
	elseif (Float64(-2.0 * x) <= 0.005)
		tmp = fma(fma(-0.3333333333333333, (x ^ 3.0), Float64(0.13333333333333333 * (x ^ 5.0))), 1.0, x);
	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], -200000.0], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.005], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + x), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -2e5

   1. Initial program 0

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

   if -2e5 < (*.f64 -2 x) < 0.0050000000000000001

   1. Initial program 58.5

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

   2. Taylor expanded in x around 0 0.5

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

   3. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, 0.13333333333333333 \cdot {x}^{5}\right), 1, x\right)} \]

   if 0.0050000000000000001 < (*.f64 -2 x)

   1. Initial program 0.0

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

   2. Taylor expanded in x around 0 1.8

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

   3. Taylor expanded in x around inf 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, 0.13333333333333333 \cdot {x}^{5}\right), 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	14024

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	0.4
Cost	7496

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -200000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 10^{-10}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.6
Cost	708

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq 0.005:\\ \;\;\;\;\frac{x \cdot 2}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Error	13.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -45197.135315993626:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.0005698361088554379:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 5
Error	13.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -45197.135315993626:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.0005698361088554379:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 6
Error	43.4
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -2.732409085953451 \cdot 10^{-308}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Error	59.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022297 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))