Average Error: 29.5 → 0.0
Time: 3.3s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\begin{array}{l} t_0 := -\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-4} + \frac{3}{x}\right) + \mathsf{fma}\left(3, {x}^{-3}, {x}^{-2}\right)\right)\right)\\ t_1 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;x \leq -3600:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2800:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t_1, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (expm1
           (log1p
            (+
             (+ (pow x -4.0) (/ 3.0 x))
             (fma 3.0 (pow x -3.0) (pow x -2.0)))))))
        (t_1 (/ (+ x -1.0) (+ x 1.0))))
   (if (<= x -3600.0)
     t_0
     (if (<= x 2800.0)
       (/ (fma x t_1 (fma -1.0 x -1.0)) (* (+ x 1.0) t_1))
       t_0))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double t_0 = -expm1(log1p(((pow(x, -4.0) + (3.0 / x)) + fma(3.0, pow(x, -3.0), pow(x, -2.0)))));
	double t_1 = (x + -1.0) / (x + 1.0);
	double tmp;
	if (x <= -3600.0) {
		tmp = t_0;
	} else if (x <= 2800.0) {
		tmp = fma(x, t_1, fma(-1.0, x, -1.0)) / ((x + 1.0) * t_1);
	} 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(-expm1(log1p(Float64(Float64((x ^ -4.0) + Float64(3.0 / x)) + fma(3.0, (x ^ -3.0), (x ^ -2.0))))))
	t_1 = Float64(Float64(x + -1.0) / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -3600.0)
		tmp = t_0;
	elseif (x <= 2800.0)
		tmp = Float64(fma(x, t_1, fma(-1.0, x, -1.0)) / Float64(Float64(x + 1.0) * t_1));
	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[(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])}, Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3600.0], t$95$0, If[LessEqual[x, 2800.0], N[(N[(x * t$95$1 + N[(-1.0 * x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
t_0 := -\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-4} + \frac{3}{x}\right) + \mathsf{fma}\left(3, {x}^{-3}, {x}^{-2}\right)\right)\right)\\
t_1 := \frac{x + -1}{x + 1}\\
\mathbf{if}\;x \leq -3600:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 2800:\\
\;\;\;\;\frac{\mathsf{fma}\left(x, t_1, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot t_1}\\

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


\end{array}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3600 or 2800 < x

    1. Initial program 59.1

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

    2. Taylor expanded in x around inf 0.3

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

    3. Applied egg-rr0.0

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

    if -3600 < x < 2800

    1. Initial program 0.1

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

    2. Applied egg-rr0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3600:\\ \;\;\;\;-\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-4} + \frac{3}{x}\right) + \mathsf{fma}\left(3, {x}^{-3}, {x}^{-2}\right)\right)\right)\\ \mathbf{elif}\;x \leq 2800:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{x + -1}{x + 1}, \mathsf{fma}\left(-1, x, -1\right)\right)}{\left(x + 1\right) \cdot \frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{-4} + \frac{3}{x}\right) + \mathsf{fma}\left(3, {x}^{-3}, {x}^{-2}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))