Average Error: 29.4 → 0.0
Time: 2.9s
Precision: binary64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
↓
\[\begin{array}{l} t_0 := {\left(e^{x}\right)}^{-2}\\ \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\log \left(e^{\mathsf{expm1}\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(t_0\right)\right)}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, \mathsf{fma}\left(0.13333333333333333, {x}^{5}, \mathsf{fma}\left(-0.05396825396825397, {x}^{7}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + t_0} + -1}\right)\\ \end{array} \]
(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (exp x) -2.0)))
   (if (<= (* -2.0 x) -0.05)
     (log (exp (expm1 (- (log1p 1.0) (log1p t_0)))))
     (if (<= (* -2.0 x) 0.01)
       (fma
        -0.3333333333333333
        (pow x 3.0)
        (fma
         0.13333333333333333
         (pow x 5.0)
         (fma -0.05396825396825397 (pow x 7.0) x)))
       (log (exp (+ (/ 2.0 (+ 1.0 t_0)) -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 t_0 = pow(exp(x), -2.0);
	double tmp;
	if ((-2.0 * x) <= -0.05) {
		tmp = log(exp(expm1((log1p(1.0) - log1p(t_0)))));
	} else if ((-2.0 * x) <= 0.01) {
		tmp = fma(-0.3333333333333333, pow(x, 3.0), fma(0.13333333333333333, pow(x, 5.0), fma(-0.05396825396825397, pow(x, 7.0), x)));
	} else {
		tmp = log(exp(((2.0 / (1.0 + t_0)) + -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)
	t_0 = exp(x) ^ -2.0
	tmp = 0.0
	if (Float64(-2.0 * x) <= -0.05)
		tmp = log(exp(expm1(Float64(log1p(1.0) - log1p(t_0)))));
	elseif (Float64(-2.0 * x) <= 0.01)
		tmp = fma(-0.3333333333333333, (x ^ 3.0), fma(0.13333333333333333, (x ^ 5.0), fma(-0.05396825396825397, (x ^ 7.0), x)));
	else
		tmp = log(exp(Float64(Float64(2.0 / Float64(1.0 + t_0)) + -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_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]}, If[LessEqual[N[(-2.0 * x), $MachinePrecision], -0.05], N[Log[N[Exp[N[(Exp[N[(N[Log[1 + 1.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.01], N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision] + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision] + N[(-0.05396825396825397 * N[Power[x, 7.0], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
t_0 := {\left(e^{x}\right)}^{-2}\\
\mathbf{if}\;-2 \cdot x \leq -0.05:\\
\;\;\;\;\log \left(e^{\mathsf{expm1}\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left(t_0\right)\right)}\right)\\

\mathbf{elif}\;-2 \cdot x \leq 0.01:\\
\;\;\;\;\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, \mathsf{fma}\left(0.13333333333333333, {x}^{5}, \mathsf{fma}\left(-0.05396825396825397, {x}^{7}, x\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\frac{2}{1 + t_0} + -1}\right)\\


\end{array}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 -2 x) < -0.050000000000000003

    1. Initial program 0.0

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

    2. Applied egg-rr0.0

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

    3. Applied egg-rr0.0

      \[\leadsto \log \left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)}}\right) \]

    if -0.050000000000000003 < (*.f64 -2 x) < 0.0100000000000000002

    1. Initial program 58.9

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

    2. Taylor expanded in x around 0 0.0

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

    3. Simplified0.0

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

    if 0.0100000000000000002 < (*.f64 -2 x)

    1. Initial program 0.0

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

    2. Applied egg-rr0.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{2}{1 + {\left(e^{x}\right)}^{-2}} + -1}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.05:\\ \;\;\;\;\log \left(e^{\mathsf{expm1}\left(\mathsf{log1p}\left(1\right) - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)}\right)\\ \mathbf{elif}\;-2 \cdot x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, {x}^{3}, \mathsf{fma}\left(0.13333333333333333, {x}^{5}, \mathsf{fma}\left(-0.05396825396825397, {x}^{7}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{2}{1 + {\left(e^{x}\right)}^{-2}} + -1}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))