Average Error: 29.4 → 0.2
Time: 4.0s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{x + 1}{x - 1}\\ \mathbf{if}\;t_0 - t_1 \leq 0.0005411699590934793:\\ \;\;\;\;\frac{-3}{x} - \left(\frac{1}{{x}^{4}} + \mathsf{fma}\left(3, {x}^{-3}, \sqrt{{x}^{-4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{t_0}\right) - t_1\\ \end{array} \]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
t_0 := \frac{x}{x + 1}\\
t_1 := \frac{x + 1}{x - 1}\\
\mathbf{if}\;t_0 - t_1 \leq 0.0005411699590934793:\\
\;\;\;\;\frac{-3}{x} - \left(\frac{1}{{x}^{4}} + \mathsf{fma}\left(3, {x}^{-3}, \sqrt{{x}^{-4}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{t_0}\right) - t_1\\


\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 (/ (+ x 1.0) (- x 1.0))))
   (if (<= (- t_0 t_1) 0.0005411699590934793)
     (-
      (/ -3.0 x)
      (+ (/ 1.0 (pow x 4.0)) (fma 3.0 (pow x -3.0) (sqrt (pow x -4.0)))))
     (- (log (exp t_0)) t_1))))
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 = (x + 1.0) / (x - 1.0);
	double tmp;
	if ((t_0 - t_1) <= 0.0005411699590934793) {
		tmp = (-3.0 / x) - ((1.0 / pow(x, 4.0)) + fma(3.0, pow(x, -3.0), sqrt(pow(x, -4.0))));
	} else {
		tmp = log(exp(t_0)) - t_1;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 5.41169959093479e-4

    1. Initial program 58.8

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.3

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

    5. Applied associate--l-_binary640.3

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

    6. Simplified0.3

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

    8. Applied add-sqr-sqrt_binary640.3

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

    9. Applied sqrt-unprod_binary640.3

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

    10. Simplified0.3

      \[\leadsto \frac{-3}{x} - \left(\frac{1}{{x}^{4}} + \mathsf{fma}\left(3, {x}^{-3}, \sqrt{\color{blue}{{x}^{-4}}}\right)\right) \]

    if 5.41169959093479e-4 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

    1. Initial program 0.0

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

    3. Applied add-log-exp_binary640.0

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

    4. Simplified0.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 0.0005411699590934793:\\ \;\;\;\;\frac{-3}{x} - \left(\frac{1}{{x}^{4}} + \mathsf{fma}\left(3, {x}^{-3}, \sqrt{{x}^{-4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{x + 1}}\right) - \frac{x + 1}{x - 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))