Average Error: 29.5 → 0.0
Time: 4.7s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\begin{array}{l} t_0 := \left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \mathbf{if}\;x \leq -66059582.89406409:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8162611635593491:\\ \;\;\;\;\frac{-1 + x \cdot -3}{\mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
t_0 := \left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\
\mathbf{if}\;x \leq -66059582.89406409:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 8162611635593491:\\
\;\;\;\;\frac{-1 + x \cdot -3}{\mathsf{fma}\left(x, x, -1\right)}\\

\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
         (-
          (- (/ -3.0 x) (+ (/ 1.0 (* x x)) (/ 3.0 (pow x 3.0))))
          (/ 1.0 (pow x 4.0)))))
   (if (<= x -66059582.89406409)
     t_0
     (if (<= x 8162611635593491.0)
       (/ (+ -1.0 (* x -3.0)) (fma x x -1.0))
       t_0))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double t_0 = ((-3.0 / x) - ((1.0 / (x * x)) + (3.0 / pow(x, 3.0)))) - (1.0 / pow(x, 4.0));
	double tmp;
	if (x <= -66059582.89406409) {
		tmp = t_0;
	} else if (x <= 8162611635593491.0) {
		tmp = (-1.0 + (x * -3.0)) / fma(x, x, -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -66059582.8940640911 or 8162611635593491 < x

    1. Initial program 60.1

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

    2. Taylor expanded in x around inf 0.3

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

    3. Simplified0.0

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

    if -66059582.8940640911 < x < 8162611635593491

    1. Initial program 0.8

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

    2. Applied frac-sub_binary640.7

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

    3. Simplified0.7

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

    4. Simplified0.7

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

    5. Taylor expanded in x around 0 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -66059582.89406409:\\ \;\;\;\;\left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \mathbf{elif}\;x \leq 8162611635593491:\\ \;\;\;\;\frac{-1 + x \cdot -3}{\mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\right) - \frac{1}{{x}^{4}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022121 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))