Asymptote C

Average Error: 29.5 → 0.2

Time: 37.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.00639665183005644 \lor \neg \left(x \le 13089.4086349961854\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(x \cdot x - 1 \cdot 1\right)\right) \cdot \left(\sqrt{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)} \cdot \left(x - 1\right)\right) - \left(\sqrt{x + 1} \cdot \left(x + 1\right)\right) \cdot \left(\sqrt{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x - 1 \cdot 1\right)\right)}{\left(\sqrt{x + 1} \cdot \left(x + 1\right)\right) \cdot \left(\sqrt{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)} \cdot \left(x - 1\right)\right)}}{\sqrt{x + 1} \cdot \left(x - 1\right)}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1.00639665183005644 or 13089.4086349961854 < x

   1. Initial program 59.1

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

   2. Taylor expanded around inf 0.6

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

   3. Simplified 0.3

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

   if -1.00639665183005644 < x < 13089.4086349961854

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

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

   4. Applied associate-/r* 0.1

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

   6. Applied frac-sub 0.1

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

   8. Applied flip-+ 0.1

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

   9. Applied flip3-+ 0.1

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

   10. Applied sqrt-div 0.1

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

   11. Applied frac-times 0.1

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

   12. Applied flip-- 0.1

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

   13. Applied frac-times 0.1

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

   14. Applied frac-sub 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.00639665183005644 \lor \neg \left(x \le 13089.4086349961854\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(x \cdot x - 1 \cdot 1\right)\right) \cdot \left(\sqrt{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)} \cdot \left(x - 1\right)\right) - \left(\sqrt{x + 1} \cdot \left(x + 1\right)\right) \cdot \left(\sqrt{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x - 1 \cdot 1\right)\right)}{\left(\sqrt{x + 1} \cdot \left(x + 1\right)\right) \cdot \left(\sqrt{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)} \cdot \left(x - 1\right)\right)}}{\sqrt{x + 1} \cdot \left(x - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))