Average Error: 28.7 → 0.0
Time: 2.9s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.86080339333128952 \cdot 10^{33} \lor \neg \left(x \le 1208365.1148162424\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-3\right) - 1}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -8.86080339333128952e33 or 1208365.1148162424 < x

    1. Initial program 59.8

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

    3. Applied add-cbrt-cube61.1

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

    4. Applied add-cbrt-cube62.7

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

    5. Applied cbrt-undiv62.7

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

    6. Simplified59.8

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

    7. Taylor expanded around inf 0.3

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

    8. Simplified0.0

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

    if -8.86080339333128952e33 < x < 1208365.1148162424

    1. Initial program 2.2

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

    3. Applied frac-sub2.2

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

    4. Simplified2.2

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

    5. Taylor expanded around 0 0.0

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

    6. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.86080339333128952 \cdot 10^{33} \lor \neg \left(x \le 1208365.1148162424\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-3\right) - 1}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))