Average Error: 29.7 → 0.1
Time: 41.0s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -10352.2765048927 \lor \neg \left(x \leq 11950.400014151677\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + {x}^{3}\right) \cdot \left(\left(x + 1\right) \cdot \left(\frac{-1 - x}{-1 + {x}^{3}} \cdot \left(-1 + x \cdot x\right)\right) + \left(x \cdot x - x\right)\right) + \left(x \cdot x\right) \cdot \left(\left(-1 - x\right) \cdot \left(-1 + x \cdot x\right)\right)}{\left(-1 + {x}^{3}\right) \cdot \left(-1 + x \cdot x\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -10352.2765048927 \lor \neg \left(x \leq 11950.400014151677\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -10352.2765048927) (not (<= x 11950.400014151677)))
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (/
    (+
     (*
      (+ -1.0 (pow x 3.0))
      (+
       (* (+ x 1.0) (* (/ (- -1.0 x) (+ -1.0 (pow x 3.0))) (+ -1.0 (* x x))))
       (- (* x x) x)))
     (* (* x x) (* (- -1.0 x) (+ -1.0 (* x x)))))
    (* (+ -1.0 (pow x 3.0)) (+ -1.0 (* x x))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -10352.2765048927) || !(x <= 11950.400014151677)) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = (((-1.0 + pow(x, 3.0)) * (((x + 1.0) * (((-1.0 - x) / (-1.0 + pow(x, 3.0))) * (-1.0 + (x * x)))) + ((x * x) - x))) + ((x * x) * ((-1.0 - x) * (-1.0 + (x * x))))) / ((-1.0 + pow(x, 3.0)) * (-1.0 + (x * x)));
	}
	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 < -10352.276504892699 or 11950.400014151677 < 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

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

    if -10352.276504892699 < x < 11950.400014151677

    1. Initial program 0.1

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

    3. Applied flip3--_binary64_14460.1

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

    4. Applied associate-/r/_binary64_13880.1

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

    5. Applied cancel-sign-sub-inv_binary64_14080.1

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

    6. Simplified0.1

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

    8. Applied distribute-rgt-in_binary64_13920.1

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

    9. Applied associate-+r+_binary64_13740.1

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

    10. Simplified0.1

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

    12. Applied associate-*r/_binary64_13840.1

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

    13. Applied flip-+_binary64_14160.1

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

    14. Applied associate-*r/_binary64_13840.1

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

    15. Applied frac-add_binary64_14500.1

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

    16. Applied frac-add_binary64_14500.1

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

    17. Simplified0.1

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

    18. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10352.2765048927 \lor \neg \left(x \leq 11950.400014151677\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-1 + {x}^{3}\right) \cdot \left(\left(x + 1\right) \cdot \left(\frac{-1 - x}{-1 + {x}^{3}} \cdot \left(-1 + x \cdot x\right)\right) + \left(x \cdot x - x\right)\right) + \left(x \cdot x\right) \cdot \left(\left(-1 - x\right) \cdot \left(-1 + x \cdot x\right)\right)}{\left(-1 + {x}^{3}\right) \cdot \left(-1 + x \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))