Average Error: 29.3 → 0.2
Time: 3.7s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \mathsf{fma}\left(x, -0.5, 1\right)\right)\\ \end{array} \]
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -0.2)
   (expm1 (- (log 2.0) (log1p (pow (exp x) -2.0))))
   (expm1 (* x (fma x -0.5 1.0)))))
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 * x) <= -0.2) {
		tmp = expm1((log(2.0) - log1p(pow(exp(x), -2.0))));
	} else {
		tmp = expm1((x * fma(x, -0.5, 1.0)));
	}
	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 * x) <= -0.2)
		tmp = expm1(Float64(log(2.0) - log1p((exp(x) ^ -2.0))));
	else
		tmp = expm1(Float64(x * fma(x, -0.5, 1.0)));
	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 * x), $MachinePrecision], -0.2], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(Exp[N[(x * N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]]

\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \leq -0.2:\\
\;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(x \cdot \mathsf{fma}\left(x, -0.5, 1\right)\right)\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 -2 x) < -0.20000000000000001

    1. Initial program 0.0

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

    2. Applied egg-rr0.0

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

    if -0.20000000000000001 < (*.f64 -2 x)

    1. Initial program 39.2

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

    2. Applied egg-rr39.1

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

    3. Taylor expanded in x around 0 0.2

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

    4. Simplified0.2

      \[\leadsto \mathsf{expm1}\left(\color{blue}{x \cdot \mathsf{fma}\left(x, -0.5, 1\right)}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.2:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x \cdot \mathsf{fma}\left(x, -0.5, 1\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))