Logistic function from Lakshay Garg

Average Error: 29.4 → 0.4

Time: 2.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{if}\;-2 \cdot x \leq -1075030.39031869:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 2.7684265819523093 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (expm1 (- (log 2.0) (log1p (pow (exp -2.0) x))))))
   (if (<= (* -2.0 x) -1075030.39031869)
     t_0
     (if (<= (* -2.0 x) 2.7684265819523093e-10)
       (fma (pow x 3.0) -0.3333333333333333 x)
       t_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 t_0 = expm1((log(2.0) - log1p(pow(exp(-2.0), x))));
	double tmp;
	if ((-2.0 * x) <= -1075030.39031869) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 2.7684265819523093e-10) {
		tmp = fma(pow(x, 3.0), -0.3333333333333333, x);
	} else {
		tmp = t_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)
	t_0 = expm1(Float64(log(2.0) - log1p((exp(-2.0) ^ x))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -1075030.39031869)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 2.7684265819523093e-10)
		tmp = fma((x ^ 3.0), -0.3333333333333333, x);
	else
		tmp = t_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_] := Block[{t$95$0 = 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], -1075030.39031869], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 2.7684265819523093e-10], N[(N[Power[x, 3.0], $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -1075030.3903186901 or 2.76842658195230929e-10 < (*.f64 -2 x)

   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 -1075030.3903186901 < (*.f64 -2 x) < 2.76842658195230929e-10

   1. Initial program 58.8

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

   2. Taylor expanded in x around 0 0.7

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

   3. Simplified 0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -1075030.39031869:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \mathbf{elif}\;-2 \cdot x \leq 2.7684265819523093 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))