Average Error: 28.6 → 0.1
Time: 4.1m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12263.528300005722 \lor \neg \left(x \le 10678.782375677225\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(x \cdot \left(\frac{1}{1 + x}\right) + \left(-\sqrt[3]{\frac{{\left(1 + x\right)}^{3}}{{\left(-1 + x\right)}^{3}}}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -12263.528300005722 or 10678.782375677225 < x

    1. Initial program 59.2

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

    3. Applied div-inv59.4

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

    4. Applied fma-neg60.2

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

    5. Taylor expanded around inf 0.3

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

    6. 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 -12263.528300005722 < x < 10678.782375677225

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    4. Applied fma-neg0.1

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

    6. Applied div-inv0.1

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

    8. Applied add-cbrt-cube0.1

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

    9. Applied add-cbrt-cube0.1

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

    10. Applied cbrt-unprod0.1

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

    11. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12263.528300005722 \lor \neg \left(x \le 10678.782375677225\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(x \cdot \left(\frac{1}{1 + x}\right) + \left(-\sqrt[3]{\frac{{\left(1 + x\right)}^{3}}{{\left(-1 + x\right)}^{3}}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 4.1m)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))