Average Error: 29.3 → 0.1
Time: 3.6m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\left(-\frac{18}{x}\right) - \frac{51}{x \cdot x}\right) - \frac{\frac{348}{x}}{x \cdot x}}{\frac{x}{1 + x} + \frac{1 + x}{x - 1}}}{\frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} \cdot \frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} + \left({\left(\frac{1 + x}{x - 1}\right)}^{\left(3 + 1\right)} + {\left(\frac{x}{1 + x}\right)}^{\left(3 + 1\right)}\right)} \le -24.6314957393821:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{1 + x}{x - 1}}\right)\\ \mathbf{if}\;\frac{\frac{\left(\left(-\frac{18}{x}\right) - \frac{51}{x \cdot x}\right) - \frac{\frac{348}{x}}{x \cdot x}}{\frac{x}{1 + x} + \frac{1 + x}{x - 1}}}{\frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} \cdot \frac{\frac{x}{1 + x}}{\frac{x - 1}{1 + x}} + \left({\left(\frac{1 + x}{x - 1}\right)}^{\left(3 + 1\right)} + {\left(\frac{x}{1 + x}\right)}^{\left(3 + 1\right)}\right)} \le 0.00035573910304800973:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3} \cdot {\left(\frac{1 + x}{x - 1}\right)}^{3}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ (/ (- (- (- (/ 18 x)) (/ 51 (* x x))) (/ (/ 348 x) (* x x))) (+ (/ x (+ 1 x)) (/ (+ 1 x) (- x 1)))) (+ (* (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x))) (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x)))) (+ (pow (/ (+ 1 x) (- x 1)) (+ 3 1)) (pow (/ x (+ 1 x)) (+ 3 1))))) < -24.6314957393821

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Applied simplify0.1

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

    if -24.6314957393821 < (/ (/ (- (- (- (/ 18 x)) (/ 51 (* x x))) (/ (/ 348 x) (* x x))) (+ (/ x (+ 1 x)) (/ (+ 1 x) (- x 1)))) (+ (* (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x))) (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x)))) (+ (pow (/ (+ 1 x) (- x 1)) (+ 3 1)) (pow (/ x (+ 1 x)) (+ 3 1))))) < 0.00035573910304800973

    1. Initial program 58.9

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

    2. Taylor expanded around inf 0.5

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

    3. Applied simplify0.2

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}}\]

    if 0.00035573910304800973 < (/ (/ (- (- (- (/ 18 x)) (/ 51 (* x x))) (/ (/ 348 x) (* x x))) (+ (/ x (+ 1 x)) (/ (+ 1 x) (- x 1)))) (+ (* (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x))) (/ (/ x (+ 1 x)) (/ (- x 1) (+ 1 x)))) (+ (pow (/ (+ 1 x) (- x 1)) (+ 3 1)) (pow (/ x (+ 1 x)) (+ 3 1)))))

    1. Initial program 0.0

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

    3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube0.1

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

    6. Applied add-cbrt-cube0.1

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

    7. Applied cbrt-undiv0.1

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

    8. Applied add-cbrt-cube0.1

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

    9. Applied add-cbrt-cube0.1

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

    10. Applied cbrt-undiv0.1

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

    11. Applied cbrt-unprod0.1

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

    12. Applied simplify0.1

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

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018198 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))