Average Error: 28.9 → 0.2
Time: 2.7s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \mathbf{if}\;t_0 \leq 0.0001:\\ \;\;\;\;-\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-4} + \frac{3}{x}\right) + \mathsf{fma}\left(3, {x}^{-3}, {x}^{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ x (+ x 1.0)) (/ (- -1.0 x) (+ x -1.0)))))
   (if (<= t_0 0.0001)
     (-
      (expm1
       (log1p
        (+ (+ (pow x -4.0) (/ 3.0 x)) (fma 3.0 (pow x -3.0) (pow x -2.0))))))
     t_0)))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double t_0 = (x / (x + 1.0)) + ((-1.0 - x) / (x + -1.0));
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = -expm1(log1p(((pow(x, -4.0) + (3.0 / x)) + fma(3.0, pow(x, -3.0), pow(x, -2.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(x)
	return Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0)))
end
↓
function code(x)
	t_0 = Float64(Float64(x / Float64(x + 1.0)) + Float64(Float64(-1.0 - x) / Float64(x + -1.0)))
	tmp = 0.0
	if (t_0 <= 0.0001)
		tmp = Float64(-expm1(log1p(Float64(Float64((x ^ -4.0) + Float64(3.0 / x)) + fma(3.0, (x ^ -3.0), (x ^ -2.0))))));
	else
		tmp = t_0;
	end
	return tmp
end

code[x_] := N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0001], (-N[(Exp[N[Log[1 + N[(N[(N[Power[x, -4.0], $MachinePrecision] + N[(3.0 / x), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[Power[x, -3.0], $MachinePrecision] + N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), t$95$0]]

\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
t_0 := \frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\
\mathbf{if}\;t_0 \leq 0.0001:\\
\;\;\;\;-\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-4} + \frac{3}{x}\right) + \mathsf{fma}\left(3, {x}^{-3}, {x}^{-2}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 1.00000000000000005e-4

    1. Initial program 58.8

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]

    2. Taylor expanded in x around inf 0.6

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{4}} + \left(3 \cdot \frac{1}{x} + \left(3 \cdot \frac{1}{{x}^{3}} + \frac{1}{{x}^{2}}\right)\right)\right)} \]

    3. Applied egg-rr0.3

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-4} + \frac{3}{x}\right) + \mathsf{fma}\left(3, {x}^{-3}, {x}^{-2}\right)\right)\right)} \]

    if 1.00000000000000005e-4 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

    1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1} \leq 0.0001:\\ \;\;\;\;-\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-4} + \frac{3}{x}\right) + \mathsf{fma}\left(3, {x}^{-3}, {x}^{-2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{-1 - x}{x + -1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))