Asymptote C

Average Error: 28.8 → 0.3

Time: 4.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -18521.5830280754635 \lor \neg \left(x \le 1.0091207646539639\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - 3 \cdot \left(\frac{1}{x} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{2 \cdot \log \left(\sqrt[3]{\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}}\right)} \cdot \sqrt[3]{e^{\log \left(\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}\right)}}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -18521.5830280754635 or 1.0091207646539639 < x

   1. Initial program 58.8

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

   2. Taylor expanded around inf 0.5

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

   3. Simplified 0.5

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

   if -18521.5830280754635 < x < 1.0091207646539639

   1. Initial program 0.0

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

   3. Applied clear-num 0.0

      \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{1}{\frac{x - 1}{x + 1}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 0.1

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

   7. Applied add-exp-log 0.1

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

   8. Applied add-exp-log 0.1

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

   9. Applied prod-exp 0.1

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

   10. Simplified 0.1

       \[\leadsto e^{\color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}}\right)}} \cdot \sqrt[3]{\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}}\]
   11. Using strategy rm

   12. Applied add-exp-log 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -18521.5830280754635 \lor \neg \left(x \le 1.0091207646539639\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - 3 \cdot \left(\frac{1}{x} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{2 \cdot \log \left(\sqrt[3]{\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}}\right)} \cdot \sqrt[3]{e^{\log \left(\frac{x}{x + 1} - \frac{1}{\frac{x - 1}{x + 1}}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))