Average Error: 29.4 → 0.4
Time: 6.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 0.0:\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 0.0:\\
\;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\\

\end{array}
double code(double x) {
	return ((double) (((double) (x / ((double) (x + 1.0)))) - ((double) (((double) (x + 1.0)) / ((double) (x - 1.0))))));
}
↓
double code(double x) {
	double VAR;
	if ((((double) (((double) (x / ((double) (x + 1.0)))) - ((double) (((double) (x + 1.0)) / ((double) (x - 1.0)))))) <= 0.0)) {
		VAR = ((double) (((double) -(((double) (((double) (1.0 / ((double) pow(x, 2.0)))) + ((double) (3.0 / x)))))) - ((double) (3.0 * ((double) (1.0 / ((double) pow(x, 3.0))))))));
	} else {
		VAR = ((double) (((double) sqrt(((double) (((double) (x / ((double) (x + 1.0)))) - ((double) (((double) (x + 1.0)) / ((double) (x - 1.0)))))))) * ((double) sqrt(((double) (((double) (x / ((double) (x + 1.0)))) - ((double) (((double) (x + 1.0)) / ((double) (x - 1.0))))))))));
	}
	return VAR;
}

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 (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) < 0.0

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)}\]
    4. Using strategy rm

    5. Applied fma-udef0.5

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

    6. Applied associate--r+0.5

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

    7. Simplified0.2

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

    if 0.0 < (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt0.7

      \[\leadsto \color{blue}{\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 0.0:\\ \;\;\;\;\left(-\left(\frac{1}{{x}^{2}} + \frac{3}{x}\right)\right) - 3 \cdot \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))