Logistic function from Lakshay Garg

Average Error: 29.5 → 0.5

Time: 5.9s

Precision: binary64

Cost: 33416

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -20000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \mathsf{fma}\left(x, 1, \mathsf{fma}\left(-0.3333333333333333, {x}^{3}, 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) -1.0)))
   (if (<= (* -2.0 x) -20000000000.0)
     t_0
     (if (<= (* -2.0 x) 0.01)
       (+
        (* -0.05396825396825397 (pow x 7.0))
        (fma
         x
         1.0
         (fma
          -0.3333333333333333
          (pow x 3.0)
          (* 0.13333333333333333 (pow x 5.0)))))
       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 = (2.0 / (1.0 + exp((-2.0 * x)))) + -1.0;
	double tmp;
	if ((-2.0 * x) <= -20000000000.0) {
		tmp = t_0;
	} else if ((-2.0 * x) <= 0.01) {
		tmp = (-0.05396825396825397 * pow(x, 7.0)) + fma(x, 1.0, fma(-0.3333333333333333, pow(x, 3.0), (0.13333333333333333 * pow(x, 5.0))));
	} 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 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) + -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -20000000000.0)
		tmp = t_0;
	elseif (Float64(-2.0 * x) <= 0.01)
		tmp = Float64(Float64(-0.05396825396825397 * (x ^ 7.0)) + fma(x, 1.0, fma(-0.3333333333333333, (x ^ 3.0), Float64(0.13333333333333333 * (x ^ 5.0)))));
	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[(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], -20000000000.0], t$95$0, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], N[(N[(-0.05396825396825397 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(x * 1.0 + N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -2e10 or 0.0100000000000000002 < (*.f64 -2 x)

   1. Initial program 0.0

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

   if -2e10 < (*.f64 -2 x) < 0.0100000000000000002

   1. Initial program 58.0

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

   2. Taylor expanded in x around 0 0.9

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

   3. Applied egg-rr 0.9

      \[\leadsto -0.05396825396825397 \cdot {x}^{7} + \color{blue}{\mathsf{fma}\left(x, 1, \mathsf{fma}\left(-0.3333333333333333, {x}^{3}, 0.13333333333333333 \cdot {x}^{5}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -20000000000:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \mathsf{fma}\left(x, 1, \mathsf{fma}\left(-0.3333333333333333, {x}^{3}, 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	20744

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -20000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.05396825396825397 \cdot {x}^{7} + \left(-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.0
Cost	14024

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	7496

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} + -1\\ \mathbf{if}\;-2 \cdot x \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;-2 \cdot x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	13.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1137000381017472:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.005868124218108132:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2 + \frac{-4}{x}\\ \end{array} \]

Alternative 5
Error	13.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1137000381017472:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{x + 2}\\ \end{array} \]

Alternative 6
Error	13.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1137000381017472:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.005868124218108132:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7
Error	43.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3819932965772264 \cdot 10^{-304}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 8
Error	46.5
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))