Average Error: 29.6 → 0.1
Time: 4.4m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{{\left((\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\right)}^{3}} \le -1.884855241363672 \cdot 10^{-05}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{(\left(\frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*}\\ \mathbf{if}\;\sqrt[3]{{\left((\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\right)}^{3}} \le 8.807831496277167 \cdot 10^{-07}:\\ \;\;\;\;\log_* (1 + \frac{\frac{\frac{7}{2}}{x} - 3}{x})\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (cbrt (pow (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) 3)) < -1.884855241363672e-05

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Applied simplify0.1

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

    if -1.884855241363672e-05 < (cbrt (pow (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) 3)) < 8.807831496277167e-07

    1. Initial program 59.6

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

    3. Applied add-log-exp59.6

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

    4. Applied add-log-exp59.6

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

    5. Applied diff-log59.6

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

    6. Applied simplify59.6

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

    7. Taylor expanded around inf 59.6

      \[\leadsto \log \color{blue}{\left(\left(1 + \frac{7}{2} \cdot \frac{1}{{x}^{2}}\right) - 3 \cdot \frac{1}{x}\right)}\]

    8. Applied simplify0.1

      \[\leadsto \color{blue}{\log_* (1 + \frac{\frac{\frac{7}{2}}{x} - 3}{x})}\]

    if 8.807831496277167e-07 < (cbrt (pow (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) 3))

    1. Initial program 0.2

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

    3. Applied flip--0.2

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed 2018167 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))