Average Error: 29.8 → 0.5
Time: 3.8s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \mathbf{if}\;t_0 \leq 1.532107773982716 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3}{x} - {x}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
t_0 := \frac{x}{x + 1} - \frac{x + 1}{x - 1}\\
\mathbf{if}\;t_0 \leq 1.532107773982716 \cdot 10^{-14}:\\
\;\;\;\;\frac{-3}{x} - {x}^{-2}\\

\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)) (/ (+ x 1.0) (- x 1.0)))))
   (if (<= t_0 1.532107773982716e-14) (- (/ -3.0 x) (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)) - ((x + 1.0) / (x - 1.0));
	double tmp;
	if (t_0 <= 1.532107773982716e-14) {
		tmp = (-3.0 / x) - pow(x, -2.0);
	} else {
		tmp = t_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 (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 1.53211e-14

    1. Initial program 59.2

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

    2. Taylor expanded in x around inf 0.7

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

    3. Simplified0.4

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1}{x \cdot x}} \]

    4. Applied pow2_binary640.4

      \[\leadsto \frac{-3}{x} - \frac{1}{\color{blue}{{x}^{2}}} \]

    5. Applied pow-flip_binary640.4

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

    6. Simplified0.4

      \[\leadsto \frac{-3}{x} - {x}^{\color{blue}{-2}} \]

    if 1.53211e-14 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

    1. Initial program 0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 1.532107773982716 \cdot 10^{-14}:\\ \;\;\;\;\frac{-3}{x} - {x}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))