Asymptote C

Average Error: 29.6 → 0.1

Time: 2.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8144.0477705834073 \lor \neg \left(x \le 6328.0624359053945\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{x + 1}{\sqrt[3]{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -8144.0477705834073 or 6328.0624359053945 < x

   1. Initial program 59.3

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

   2. Taylor expanded around inf 0.3

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

   3. Simplified 0.0

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

   if -8144.0477705834073 < x < 6328.0624359053945

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.1

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

   4. Applied *-un-lft-identity 0.1

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

   5. Applied times-frac 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8144.0477705834073 \lor \neg \left(x \le 6328.0624359053945\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{1}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}} \cdot \frac{x + 1}{\sqrt[3]{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))