Asymptote C

Average Error: 29.2 → 0.2

Time: 6.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.993340841827823673 \lor \neg \left(x \le 13973.283901258525\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{3} + \left(x \cdot \left(1 \cdot 1 - x \cdot 1\right) - \frac{{x}^{3} + {1}^{3}}{x - 1} \cdot \left(x + 1\right)\right)}{\sqrt{\left(1 \cdot \left(1 - x\right) + {x}^{2}\right) \cdot \left(x + 1\right)} \cdot \sqrt{\left(1 \cdot \left(1 - x\right) + {x}^{2}\right) \cdot \left(x + 1\right)}}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -0.9933408418278237 or 13973.283901258525 < x

   1. Initial program 58.7

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

   3. Applied add-exp-log 59.4

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

   4. Taylor expanded around inf 0.6

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

   5. Simplified 0.2

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

   if -0.9933408418278237 < x < 13973.283901258525

   1. Initial program 0.1

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

   3. Applied div-inv 0.1

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

   5. Applied flip3-+ 0.1

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

   6. Applied associate-*l/ 0.1

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

   7. Applied frac-sub 0.1

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

   8. Simplified 0.1

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

   9. Simplified 0.1

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

   11. Applied add-sqr-sqrt 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.993340841827823673 \lor \neg \left(x \le 13973.283901258525\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{3} + \left(x \cdot \left(1 \cdot 1 - x \cdot 1\right) - \frac{{x}^{3} + {1}^{3}}{x - 1} \cdot \left(x + 1\right)\right)}{\sqrt{\left(1 \cdot \left(1 - x\right) + {x}^{2}\right) \cdot \left(x + 1\right)} \cdot \sqrt{\left(1 \cdot \left(1 - x\right) + {x}^{2}\right) \cdot \left(x + 1\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))