Logistic function from Lakshay Garg

Average Error: 29.2 → 0.3

Time: 3.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ t_1 := \mathsf{fma}\left({t_0}^{-0.5}, \frac{2}{\sqrt{t_0}}, -1\right)\\ \mathbf{if}\;-2 \cdot x \leq -459137.7511145799:\\ \;\;\;\;t_1\\ \mathbf{elif}\;-2 \cdot x \leq 0.002598313578064034:\\ \;\;\;\;\mathsf{fma}\left({x}^{7}, -0.05396825396825397, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, \mathsf{fma}\left(0.13333333333333333, {x}^{5}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x))))
        (t_1 (fma (pow t_0 -0.5) (/ 2.0 (sqrt t_0)) -1.0)))
   (if (<= (* -2.0 x) -459137.7511145799)
     t_1
     (if (<= (* -2.0 x) 0.002598313578064034)
       (fma
        (pow x 7.0)
        -0.05396825396825397
        (fma
         (pow x 3.0)
         -0.3333333333333333
         (fma 0.13333333333333333 (pow x 5.0) x)))
       t_1))))

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 = 1.0 + exp((-2.0 * x));
	double t_1 = fma(pow(t_0, -0.5), (2.0 / sqrt(t_0)), -1.0);
	double tmp;
	if ((-2.0 * x) <= -459137.7511145799) {
		tmp = t_1;
	} else if ((-2.0 * x) <= 0.002598313578064034) {
		tmp = fma(pow(x, 7.0), -0.05396825396825397, fma(pow(x, 3.0), -0.3333333333333333, fma(0.13333333333333333, pow(x, 5.0), x)));
	} else {
		tmp = t_1;
	}
	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(1.0 + exp(Float64(-2.0 * x)))
	t_1 = fma((t_0 ^ -0.5), Float64(2.0 / sqrt(t_0)), -1.0)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -459137.7511145799)
		tmp = t_1;
	elseif (Float64(-2.0 * x) <= 0.002598313578064034)
		tmp = fma((x ^ 7.0), -0.05396825396825397, fma((x ^ 3.0), -0.3333333333333333, fma(0.13333333333333333, (x ^ 5.0), x)));
	else
		tmp = t_1;
	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[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, -0.5], $MachinePrecision] * N[(2.0 / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -459137.7511145799], t$95$1, If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002598313578064034], N[(N[Power[x, 7.0], $MachinePrecision] * -0.05396825396825397 + N[(N[Power[x, 3.0], $MachinePrecision] * -0.3333333333333333 + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Error

Bits error versus x

Bits error versus y

Derivation

1. Split input into 2 regimes

2. if (*.f64 -2 x) < -459137.751114579907 or 0.0025983135780640338 < (*.f64 -2 x)

   1. Initial program 0.0

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

   2. Applied add-sqr-sqrt_binary64 0.0

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

   3. Applied *-un-lft-identity_binary64 0.0

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

   4. Applied times-frac_binary64 0.0

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

   5. Applied fma-neg_binary64 0.0

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

   6. Simplified 0.0

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

   7. Applied pow1/2_binary64 0.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{{\left(1 + e^{-2 \cdot x}\right)}^{0.5}}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right) \]

   8. Applied pow-flip_binary64 0.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(1 + e^{-2 \cdot x}\right)}^{\left(-0.5\right)}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right) \]

   9. Simplified 0.0

      \[\leadsto \mathsf{fma}\left({\left(1 + e^{-2 \cdot x}\right)}^{\color{blue}{-0.5}}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right) \]

   if -459137.751114579907 < (*.f64 -2 x) < 0.0025983135780640338

   1. Initial program 58.4

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

   2. Taylor expanded in x around 0 0.5

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

   3. Simplified 0.5

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

   4. Taylor expanded in x around 0 0.5

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

   5. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -459137.7511145799:\\ \;\;\;\;\mathsf{fma}\left({\left(1 + e^{-2 \cdot x}\right)}^{-0.5}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.002598313578064034:\\ \;\;\;\;\mathsf{fma}\left({x}^{7}, -0.05396825396825397, \mathsf{fma}\left({x}^{3}, -0.3333333333333333, \mathsf{fma}\left(0.13333333333333333, {x}^{5}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(1 + e^{-2 \cdot x}\right)}^{-0.5}, \frac{2}{\sqrt{1 + e^{-2 \cdot x}}}, -1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))