Asymptote C

Average Error: 28.7 → 0.2

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 1.45135244523 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right) \cdot \sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}} \cdot \sqrt{\frac{x}{x + 1} - \sqrt[3]{{\left(\frac{x + 1}{x - 1}\right)}^{3}}}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) < 1.45135244523e-6

   1. Initial program 59.3

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

   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 0.2

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

   if 1.45135244523e-6 < (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))

   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

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

   5. Applied add-cbrt-cube 0.1

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

   6. Applied add-cbrt-cube 0.1

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

   7. Applied cbrt-undiv 0.1

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

   8. Simplified 0.1

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

   10. Applied add-cube-cbrt 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 1.45135244523 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\right) \cdot \sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}} \cdot \sqrt{\frac{x}{x + 1} - \sqrt[3]{{\left(\frac{x + 1}{x - 1}\right)}^{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))