Average Error: 29.0 → 0.1
Time: 3.3s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -9786.608325632764 \lor \neg \left(x \leq 9787.463873477589\right):\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{{x}^{3} + 1} \cdot \left(x \cdot x + \left(1 - x\right)\right) - \frac{x + 1}{x + -1}\right)}^{3}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -9786.608325632764 \lor \neg \left(x \leq 9787.463873477589\right):\\
\;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -9786.608325632764) (not (<= x 9787.463873477589)))
   (+ (- (/ -1.0 (* x x)) (/ 3.0 x)) (/ -3.0 (pow x 3.0)))
   (cbrt
    (pow
     (-
      (* (/ x (+ (pow x 3.0) 1.0)) (+ (* x x) (- 1.0 x)))
      (/ (+ x 1.0) (+ x -1.0)))
     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 <= -9786.608325632764) || !(x <= 9787.463873477589)) {
		tmp = ((-1.0 / (x * x)) - (3.0 / x)) + (-3.0 / pow(x, 3.0));
	} else {
		tmp = cbrt(pow((((x / (pow(x, 3.0) + 1.0)) * ((x * x) + (1.0 - x))) - ((x + 1.0) / (x + -1.0))), 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 2 regimes

  2. if x < -9786.60832563276381 or 9787.463873477589 < x

    1. Initial program 59.4

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

    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

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

    if -9786.60832563276381 < x < 9787.463873477589

    1. Initial program 0.1

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

    3. Applied flip3-+_binary640.1

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

    4. Applied associate-/r/_binary640.1

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

    5. Simplified0.1

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

    7. Applied add-cbrt-cube_binary640.1

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

    8. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9786.608325632764 \lor \neg \left(x \leq 9787.463873477589\right):\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x}{{x}^{3} + 1} \cdot \left(x \cdot x + \left(1 - x\right)\right) - \frac{x + 1}{x + -1}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))