Average Error: 29.4 → 0.1
Time: 5.3s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 6.660126104396369 \cdot 10^{-06}:\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1 + {x}^{3}}{\left(x - 1\right) \cdot \left(x \cdot x + \left(1 - x\right)\right)}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 6.660126104396369 \cdot 10^{-06}:\\
\;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) 6.660126104396369e-06)
   (- (- (/ -3.0 x) (pow x -2.0)) (/ 3.0 (pow x 3.0)))
   (-
    (/ x (+ x 1.0))
    (/ (+ 1.0 (pow x 3.0)) (* (- x 1.0) (+ (* x x) (- 1.0 x)))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if (((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0))) <= 6.660126104396369e-06) {
		tmp = ((-3.0 / x) - pow(x, -2.0)) - (3.0 / pow(x, 3.0));
	} else {
		tmp = (x / (x + 1.0)) - ((1.0 + pow(x, 3.0)) / ((x - 1.0) * ((x * x) + (1.0 - 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 (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))) < 6.6601261044e-6

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.2

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

    5. Applied pow2_binary64_22050.2

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

    6. Applied pow-flip_binary64_21980.2

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

    7. Simplified0.2

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

    if 6.6601261044e-6 < (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))

    1. Initial program 0.1

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

    3. Applied flip3-+_binary64_21270.1

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

    4. Applied associate-/l/_binary64_20710.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \leq 6.660126104396369 \cdot 10^{-06}:\\ \;\;\;\;\left(\frac{-3}{x} - {x}^{-2}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1 + {x}^{3}}{\left(x - 1\right) \cdot \left(x \cdot x + \left(1 - x\right)\right)}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021102 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))