Average Error: 28.8 → 0.1
Time: 5.0m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12835.933082253438 \lor \neg \left(x \le 12450.535684542745\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{(x \cdot x + -1)_*}{\left(x - 1\right) \cdot \left(x - 1\right)}}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -12835.933082253438 or 12450.535684542745 < x

    1. Initial program 59.2

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

    3. Applied add-log-exp59.2

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

    4. Taylor expanded around -inf 0.4

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

    5. Simplified0.0

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

    if -12835.933082253438 < x < 12450.535684542745

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    5. Applied flip-+0.1

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

    6. Applied associate-/l/0.1

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12835.933082253438 \lor \neg \left(x \le 12450.535684542745\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x} - \frac{(x \cdot x + -1)_*}{\left(x - 1\right) \cdot \left(x - 1\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 5.0m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.70.10.029.699.9%
herbie shell --seed 2018290 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))