Logistic function from Lakshay Garg

Average Error: 29.6 → 0.2

Time: 7.4s

Precision: binary64

Cost: 39880

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;t_0 + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, {t_0}^{0.6666666666666666}, -1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 2.0 (+ 1.0 (exp (* -2.0 x))))))
   (if (<= (* -2.0 x) -200.0)
     (+ t_0 -1.0)
     (if (<= (* -2.0 x) 0.0002)
       (+ x (* -0.3333333333333333 (pow x 3.0)))
       (fma
        (cbrt (/ 2.0 (+ 1.0 (pow (exp x) -2.0))))
        (pow t_0 0.6666666666666666)
        -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 t_0 = 2.0 / (1.0 + exp((-2.0 * x)));
	double tmp;
	if ((-2.0 * x) <= -200.0) {
		tmp = t_0 + -1.0;
	} else if ((-2.0 * x) <= 0.0002) {
		tmp = x + (-0.3333333333333333 * pow(x, 3.0));
	} else {
		tmp = fma(cbrt((2.0 / (1.0 + pow(exp(x), -2.0)))), pow(t_0, 0.6666666666666666), -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)
	t_0 = Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x))))
	tmp = 0.0
	if (Float64(-2.0 * x) <= -200.0)
		tmp = Float64(t_0 + -1.0);
	elseif (Float64(-2.0 * x) <= 0.0002)
		tmp = Float64(x + Float64(-0.3333333333333333 * (x ^ 3.0)));
	else
		tmp = fma(cbrt(Float64(2.0 / Float64(1.0 + (exp(x) ^ -2.0)))), (t_0 ^ 0.6666666666666666), -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_] := Block[{t$95$0 = N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -200.0], N[(t$95$0 + -1.0), $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.0002], N[(x + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 / N[(1.0 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$0, 0.6666666666666666], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -200

   1. Initial program 0

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

   if -200 < (*.f64 -2 x) < 2.0000000000000001e-4

   1. Initial program 58.9

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

   2. Taylor expanded in x around 0 0.3

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

   if 2.0000000000000001e-4 < (*.f64 -2 x)

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, \sqrt[3]{4 \cdot {\left(1 + {\left(e^{x}\right)}^{-2}\right)}^{-2}}, -1\right)} \]

   3. Applied egg-rr 0.1

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, \color{blue}{{\left(\frac{2}{1 + {\left(e^{x}\right)}^{-2}}\right)}^{0.6666666666666666}}, -1\right) \]

   4. Applied egg-rr 0.1

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, {\left(\frac{2}{1 + \color{blue}{e^{x \cdot -2}}}\right)}^{0.6666666666666666}, -1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{\frac{2}{1 + {\left(e^{x}\right)}^{-2}}}, {\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{0.6666666666666666}, -1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	26440

\[\begin{array}{l} t_0 := e^{-2 \cdot x}\\ \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;\frac{2}{1 + t_0} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(t_0\right)\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7496

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.0002:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -8114724.489725059:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \frac{1}{x + 2}\\ \end{array} \]

Alternative 4
Error	13.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8114724.489725059:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7.353595432915118 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 5
Error	13.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -8114724.489725059:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 7.353595432915118 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 6
Error	28.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 7.353595432915118 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Error	30.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))