Average Error: 29.3 → 0.5
Time: 3.3s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x + {x}^{2} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \end{array} \]
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (expm1 (+ x (* (pow x 2.0) -0.5)))
   (expm1 (- (log 2.0) (log1p (pow (exp x) -2.0))))))
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 / (1.0 + exp((-2.0 * x)))) <= 1.0) {
		tmp = expm1((x + (pow(x, 2.0) * -0.5)));
	} else {
		tmp = expm1((log(2.0) - log1p(pow(exp(x), -2.0))));
	}
	return tmp;
}

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}
↓
public static double code(double x, double y) {
	double tmp;
	if ((2.0 / (1.0 + Math.exp((-2.0 * x)))) <= 1.0) {
		tmp = Math.expm1((x + (Math.pow(x, 2.0) * -0.5)));
	} else {
		tmp = Math.expm1((Math.log(2.0) - Math.log1p(Math.pow(Math.exp(x), -2.0))));
	}
	return tmp;
}

def code(x, y):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0
↓
def code(x, y):
	tmp = 0
	if (2.0 / (1.0 + math.exp((-2.0 * x)))) <= 1.0:
		tmp = math.expm1((x + (math.pow(x, 2.0) * -0.5)))
	else:
		tmp = math.expm1((math.log(2.0) - math.log1p(math.pow(math.exp(x), -2.0))))
	return tmp

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 / Float64(1.0 + exp(Float64(-2.0 * x)))) <= 1.0)
		tmp = expm1(Float64(x + Float64((x ^ 2.0) * -0.5)));
	else
		tmp = expm1(Float64(log(2.0) - log1p((exp(x) ^ -2.0))));
	end
	return tmp
end

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 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(Exp[N[(x + N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]

\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 1:\\
\;\;\;\;\mathsf{expm1}\left(x + {x}^{2} \cdot -0.5\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\


\end{array}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 2 (+.f64 1 (exp.f64 (*.f64 -2 x)))) < 1

    1. Initial program 39.7

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

    2. Applied egg-rr39.6

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

    3. Taylor expanded in x around 0 0.2

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x - 0.5 \cdot {x}^{2}}\right) \]

    if 1 < (/.f64 2 (+.f64 1 (exp.f64 (*.f64 -2 x))))

    1. Initial program 1.3

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

    2. Applied egg-rr1.3

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)} \]

    3. Taylor expanded in x around inf 1.3

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right) \]

    4. Simplified1.3

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{1 + e^{-2 \cdot x}} \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x + {x}^{2} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))