Average Error: 29.3 → 0.1
Time: 2.7s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -11799.423822239964 \lor \neg \left(x \leq 10786.998680762768\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1 + {x}^{3}}{\left(x + -1\right) \cdot \left(1 + x \cdot \left(x + -1\right)\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -11799.423822239964 \lor \neg \left(x \leq 10786.998680762768\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{3}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} - \frac{1 + {x}^{3}}{\left(x + -1\right) \cdot \left(1 + x \cdot \left(x + -1\right)\right)}\\

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -11799.423822239964) (not (<= x 10786.998680762768)))
   (- (/ -1.0 (* x x)) (* (+ 1.0 (/ 1.0 (* x x))) (/ 3.0 x)))
   (-
    (/ x (+ x 1.0))
    (/ (+ 1.0 (pow x 3.0)) (* (+ x -1.0) (+ 1.0 (* x (+ x -1.0))))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -11799.423822239964) || !(x <= 10786.998680762768)) {
		tmp = (-1.0 / (x * x)) - ((1.0 + (1.0 / (x * x))) * (3.0 / x));
	} else {
		tmp = (x / (x + 1.0)) - ((1.0 + pow(x, 3.0)) / ((x + -1.0) * (1.0 + (x * (x + -1.0)))));
	}
	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 x < -11799.4238222399636 or 10786.998680762768 < x

    1. Initial program 59.3

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

    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

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

    5. Applied unpow3_binary640.0

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

    6. Applied *-un-lft-identity_binary640.0

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

    7. Applied times-frac_binary640.0

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

    8. Applied distribute-rgt1-in_binary640.0

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

    9. Simplified0.0

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

    if -11799.4238222399636 < x < 10786.998680762768

    1. Initial program 0.1

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

    3. Applied flip3-+_binary640.1

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

    4. Applied associate-/l/_binary640.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11799.423822239964 \lor \neg \left(x \leq 10786.998680762768\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(1 + \frac{1}{x \cdot x}\right) \cdot \frac{3}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1 + {x}^{3}}{\left(x + -1\right) \cdot \left(1 + x \cdot \left(x + -1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))