Average Error: 29.0 → 0.1
Time: 7.5s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -11819.493831145772 \lor \neg \left(x \leq 11868.407629700492\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \log \left(e^{\frac{x + 1}{x + -1}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -11819.493831145772 \lor \neg \left(x \leq 11868.407629700492\right):\\
\;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -11819.493831145772) (not (<= x 11868.407629700492)))
   (- (- (/ -3.0 x) (/ 1.0 (* x x))) (/ 3.0 (pow x 3.0)))
   (- (/ x (+ x 1.0)) (log (exp (/ (+ x 1.0) (+ x -1.0)))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -11819.493831145772) || !(x <= 11868.407629700492)) {
		tmp = ((-3.0 / x) - (1.0 / (x * x))) - (3.0 / pow(x, 3.0));
	} else {
		tmp = (x / (x + 1.0)) - log(exp((x + 1.0) / (x + -1.0)));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -11819.4938311457718 or 11868.4076297004922 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -11819.4938311457718 < x < 11868.4076297004922

    1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied add-log-exp_binary64_18220.1

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11819.493831145772 \lor \neg \left(x \leq 11868.407629700492\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \log \left(e^{\frac{x + 1}{x + -1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021060 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))