Average Error: 28.7 → 0.1
Time: 6.7s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := {\left(x + 1\right)}^{3}\\ t_2 := {\left(x + -1\right)}^{3}\\ t_3 := \frac{-3}{x} + \frac{\frac{-1}{x}}{x} \cdot \left(\frac{3}{x} + 1\right)\\ t_4 := \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}\\ \mathbf{if}\;x \leq -2181902.375756449:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 547.9582968275782:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, -1, -1\right)\right)}^{3}, t_1, t_2 \cdot {x}^{3}\right)}{t_1 \cdot t_2}}{{t_0}^{2} + \left({t_4}^{2} - t_0 \cdot t_4\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
(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)))
        (t_1 (pow (+ x 1.0) 3.0))
        (t_2 (pow (+ x -1.0) 3.0))
        (t_3 (+ (/ -3.0 x) (* (/ (/ -1.0 x) x) (+ (/ 3.0 x) 1.0))))
        (t_4 (/ (fma -1.0 x -1.0) (+ x -1.0))))
   (if (<= x -2181902.375756449)
     t_3
     (if (<= x 547.9582968275782)
       (/
        (/
         (fma (pow (fma x -1.0 -1.0) 3.0) t_1 (* t_2 (pow x 3.0)))
         (* t_1 t_2))
        (+ (pow t_0 2.0) (- (pow t_4 2.0) (* t_0 t_4))))
       t_3))))
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);
	double t_1 = pow((x + 1.0), 3.0);
	double t_2 = pow((x + -1.0), 3.0);
	double t_3 = (-3.0 / x) + (((-1.0 / x) / x) * ((3.0 / x) + 1.0));
	double t_4 = fma(-1.0, x, -1.0) / (x + -1.0);
	double tmp;
	if (x <= -2181902.375756449) {
		tmp = t_3;
	} else if (x <= 547.9582968275782) {
		tmp = (fma(pow(fma(x, -1.0, -1.0), 3.0), t_1, (t_2 * pow(x, 3.0))) / (t_1 * t_2)) / (pow(t_0, 2.0) + (pow(t_4, 2.0) - (t_0 * t_4)));
	} else {
		tmp = t_3;
	}
	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(x / Float64(x + 1.0))
	t_1 = Float64(x + 1.0) ^ 3.0
	t_2 = Float64(x + -1.0) ^ 3.0
	t_3 = Float64(Float64(-3.0 / x) + Float64(Float64(Float64(-1.0 / x) / x) * Float64(Float64(3.0 / x) + 1.0)))
	t_4 = Float64(fma(-1.0, x, -1.0) / Float64(x + -1.0))
	tmp = 0.0
	if (x <= -2181902.375756449)
		tmp = t_3;
	elseif (x <= 547.9582968275782)
		tmp = Float64(Float64(fma((fma(x, -1.0, -1.0) ^ 3.0), t_1, Float64(t_2 * (x ^ 3.0))) / Float64(t_1 * t_2)) / Float64((t_0 ^ 2.0) + Float64((t_4 ^ 2.0) - Float64(t_0 * t_4))));
	else
		tmp = t_3;
	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[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x + 1.0), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(x + -1.0), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-3.0 / x), $MachinePrecision] + N[(N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision] * N[(N[(3.0 / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 * x + -1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2181902.375756449], t$95$3, If[LessEqual[x, 547.9582968275782], N[(N[(N[(N[Power[N[(x * -1.0 + -1.0), $MachinePrecision], 3.0], $MachinePrecision] * t$95$1 + N[(t$95$2 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] - N[(t$95$0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
t_0 := \frac{x}{x + 1}\\
t_1 := {\left(x + 1\right)}^{3}\\
t_2 := {\left(x + -1\right)}^{3}\\
t_3 := \frac{-3}{x} + \frac{\frac{-1}{x}}{x} \cdot \left(\frac{3}{x} + 1\right)\\
t_4 := \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}\\
\mathbf{if}\;x \leq -2181902.375756449:\\
\;\;\;\;t_3\\

\mathbf{elif}\;x \leq 547.9582968275782:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, -1, -1\right)\right)}^{3}, t_1, t_2 \cdot {x}^{3}\right)}{t_1 \cdot t_2}}{{t_0}^{2} + \left({t_4}^{2} - t_0 \cdot t_4\right)}\\

\mathbf{else}:\\
\;\;\;\;t_3\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if x < -2181902.3757564491 or 547.958296827578238 < x

    1. Initial program 59.3

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

    2. Taylor expanded in x around inf 0.4

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

    3. Simplified0.1

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

    if -2181902.3757564491 < x < 547.958296827578238

    1. Initial program 0.1

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

    2. Applied egg-rr0.1

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

    3. Applied egg-rr0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2181902.375756449:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x} \cdot \left(\frac{3}{x} + 1\right)\\ \mathbf{elif}\;x \leq 547.9582968275782:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, -1, -1\right)\right)}^{3}, {\left(x + 1\right)}^{3}, {\left(x + -1\right)}^{3} \cdot {x}^{3}\right)}{{\left(x + 1\right)}^{3} \cdot {\left(x + -1\right)}^{3}}}{{\left(\frac{x}{x + 1}\right)}^{2} + \left({\left(\frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}\right)}^{2} - \frac{x}{x + 1} \cdot \frac{\mathsf{fma}\left(-1, x, -1\right)}{x + -1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} + \frac{\frac{-1}{x}}{x} \cdot \left(\frac{3}{x} + 1\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))