Average Error: 29.0 → 0.1
Time: 1.7min
Precision: binary64
Cost: 1412
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -10019.828485270371 \lor \neg \left(x \leq 14603.16566047444\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \log \left(e^{e^{\frac{x}{x + 1} - \frac{x + 1}{x + -1}}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -10019.828485270371 \lor \neg \left(x \leq 14603.16566047444\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -10019.828485270371) (not (<= x 14603.16566047444)))
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (log (log (exp (exp (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ 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 <= -10019.828485270371) || !(x <= 14603.16566047444)) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = log(log(exp(exp((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy29.0
Cost1088
\[\log \left(e^{\frac{1}{\frac{1 + x}{x}} - \frac{1 + x}{x - 1}}\right)\]
Alternative 2
Accuracy29.0
Cost1152
\[\log \left(e^{\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)}^{3}}}\right)\]
Alternative 3
Accuracy29.4
Cost1920
\[\log \left(e^{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left|\sqrt[3]{1}\right|\right) \cdot \frac{\sqrt{\sqrt[3]{1}}}{\frac{x + 1}{\sqrt[3]{x}}} - \frac{x + 1}{x - 1}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -10019.8284852703709 or 14603.1656604744403 < x

    1. Initial program 59.6

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

    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 -10019.8284852703709 < x < 14603.1656604744403

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary64_21630.1

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

    4. Applied add-log-exp_binary64_21630.1

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

    5. Applied diff-log_binary64_22160.1

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

    6. Simplified0.1

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

    8. Applied clear-num_binary64_21230.1

      \[\leadsto \log \left(e^{\color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{x + 1}{x - 1}}\right)\]
    9. Using strategy rm

    10. Applied add-log-exp_binary64_21630.1

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

    11. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10019.828485270371 \lor \neg \left(x \leq 14603.16566047444\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \log \left(e^{e^{\frac{x}{x + 1} - \frac{x + 1}{x + -1}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))