Average Error: 29.6 → 0.5
Time: 7.9m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -4.507052863629578 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{1 + x} - \log \left(e^{\frac{1 + x}{x - 1}}\right)\\ \mathbf{if}\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 3.5277422585388602 \cdot 10^{-06}:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(1 + x\right) \cdot \left(\frac{-1}{x - 1}\right) + \left(\frac{x}{1 + x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ (/ 3 x) 1) (/ (- 1) (* x x)) (/ (- 3) x)) < -4.507052863629578e-19

    1. Initial program 1.6

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

    3. Applied add-log-exp1.6

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

    if -4.507052863629578e-19 < (fma (+ (/ 3 x) 1) (/ (- 1) (* x x)) (/ (- 3) x)) < 3.5277422585388602e-06

    1. Initial program 60.0

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

    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. Applied simplify0.0

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

    if 3.5277422585388602e-06 < (fma (+ (/ 3 x) 1) (/ (- 1) (* x x)) (/ (- 3) x))

    1. Initial program 0.2

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

    3. Applied flip--0.2

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

    4. Applied associate-/r/0.2

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

    5. Applied add-cube-cbrt0.2

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

    6. Applied prod-diff0.2

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

    7. Applied simplify0.2

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

    8. Applied simplify0.2

      \[\leadsto (\left(x + 1\right) \cdot \left(\frac{-1}{x - 1}\right) + \left(\frac{x}{x + 1}\right))_* + \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify0.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -4.507052863629578 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{1 + x} - \log \left(e^{\frac{1 + x}{x - 1}}\right)\\ \mathbf{if}\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 3.5277422585388602 \cdot 10^{-06}:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(1 + x\right) \cdot \left(\frac{-1}{x - 1}\right) + \left(\frac{x}{1 + x}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 7.9m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))