Average Error: 28.9 → 0.1
Time: 8.0m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15984.259334689732 \lor \neg \left(x \le 13680.602275180321\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{1 + x}\right)}^{3} - \sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3}} \cdot \left(\sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3}}\right)}{\left(\frac{x}{1 + x} \cdot \frac{1 + x}{x - 1} + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right) + \frac{x}{1 + x} \cdot \frac{x}{1 + x}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -15984.259334689732 or 13680.602275180321 < x

    1. Initial program 59.2

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

    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. Simplified0.0

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

    if -15984.259334689732 < x < 13680.602275180321

    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-cube-cbrt0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15984.259334689732 \lor \neg \left(x \le 13680.602275180321\right):\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{1 + x}\right)}^{3} - \sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3}} \cdot \left(\sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{1 + x}{x - 1}\right)}^{3}}\right)}{\left(\frac{x}{1 + x} \cdot \frac{1 + x}{x - 1} + \frac{1 + x}{x - 1} \cdot \frac{1 + x}{x - 1}\right) + \frac{x}{1 + x} \cdot \frac{x}{1 + x}}\\ \end{array}\]

Runtime

Time bar (total: 8.0m)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))