Average Error: 29.6 → 0.1
Time: 4.5m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -14940.500313875807 \lor \neg \left(x \le 11127.418247084534\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x}}\right) - \frac{(x \cdot x + -1)_*}{\left(x - 1\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -14940.500313875807 or 11127.418247084534 < 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

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

    5. Applied flip-+60.9

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

    6. Applied associate-/l/61.0

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

    7. Simplified61.0

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

    8. Taylor expanded around inf 0.3

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

    9. 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 -14940.500313875807 < x < 11127.418247084534

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    5. Applied flip-+0.1

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

    6. Applied associate-/l/0.1

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -14940.500313875807 \lor \neg \left(x \le 11127.418247084534\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{x}{1 + x}}\right) - \frac{(x \cdot x + -1)_*}{\left(x - 1\right) \cdot \left(x - 1\right)}\\ \end{array}\]

Runtime

Time bar (total: 4.5m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes30.50.10.030.5100%
herbie shell --seed 2018355 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))