Average Error: 29.4 → 0.2
Time: 4.6m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9400.702088605463:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) \cdot 3\\ \mathbf{elif}\;x \le 12258.439414387118:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}}\right) + \log \left(\sqrt{e^{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}}\right)}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \frac{1 + x}{x - 1}\right) + \frac{x}{1 + x} \cdot \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) \cdot 3\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -9400.702088605463 or 12258.439414387118 < x

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    if -9400.702088605463 < x < 12258.439414387118

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    5. Applied add-log-exp0.2

      \[\leadsto \frac{\color{blue}{\log \left(e^{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}\right)}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \left(\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1} + \frac{x}{x + 1} \cdot \frac{x + 1}{x - 1}\right)}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.2

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

    8. Applied log-prod0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9400.702088605463:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) \cdot 3\\ \mathbf{elif}\;x \le 12258.439414387118:\\ \;\;\;\;\frac{\log \left(\sqrt{e^{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}}\right) + \log \left(\sqrt{e^{{\left(\frac{x}{1 + x}\right)}^{3} - {\left(\frac{1 + x}{x - 1}\right)}^{3}}}\right)}{\left(\frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1} + \frac{x}{1 + x} \cdot \frac{1 + x}{x - 1}\right) + \frac{x}{1 + x} \cdot \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} + \left(\frac{-1}{x} + \frac{\frac{-1}{x}}{x \cdot x}\right) \cdot 3\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))