Average Error: 29.2 → 0.0
Time: 6.8s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.660453270019914 \cdot 10^{+88}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 171369.6908751599:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{\frac{x \cdot x + -1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.660453270019914 \cdot 10^{+88}:\\
\;\;\;\;\frac{-3}{x}\\

\mathbf{elif}\;x \leq 171369.6908751599:\\
\;\;\;\;\frac{\frac{-1}{x} - 3}{\frac{x \cdot x + -1}{x}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -2.660453270019914e+88)
   (/ -3.0 x)
   (if (<= x 171369.6908751599)
     (/ (- (/ -1.0 x) 3.0) (/ (+ (* x x) -1.0) x))
     (- (- (/ -3.0 x) (pow x -2.0)) (/ 3.0 (pow x 3.0))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if (x <= -2.660453270019914e+88) {
		tmp = -3.0 / x;
	} else if (x <= 171369.6908751599) {
		tmp = ((-1.0 / x) - 3.0) / (((x * x) + -1.0) / x);
	} else {
		tmp = ((-3.0 / x) - pow(x, -2.0)) - (3.0 / pow(x, 3.0));
	}
	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 3 regimes

  2. if x < -2.660453270019914e88

    1. Initial program 60.1

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

    2. Taylor expanded around inf 0

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

    if -2.660453270019914e88 < x < 171369.690875159897

    1. Initial program 7.3

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

    3. Applied clear-num_binary64_38287.3

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

    4. Simplified7.3

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + x}{x}}} - \frac{x + 1}{x - 1}\]
    5. Using strategy rm

    6. Applied frac-sub_binary64_38387.0

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

    7. Simplified7.0

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

    8. Simplified7.0

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

    9. Taylor expanded around 0 0.0

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

    if 171369.690875159897 < x

    1. Initial program 59.7

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

    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

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

    5. Applied pow2_binary64_39100.0

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

    6. Applied pow-flip_binary64_39030.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.660453270019914 \cdot 10^{+88}:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 171369.6908751599:\\ \;\;\;\;\frac{\frac{-1}{x} - 3}{\frac{x \cdot x + -1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))