Average Error: 29.9 → 0.3
Time: 37.7s
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le -3.456878228130283 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x + 1}}}{\sqrt{x + 1}} - \frac{x + 1}{x - 1}\\ \mathbf{if}\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x} \le 6.019744398695255 \cdot 10^{-10}:\\ \;\;\;\;\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{x}{x + 1} - \frac{x + 1}{x - 1}\right)}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < -3.456878228130283e-10

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt0.6

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

    4. Applied associate-/r*0.6

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

    if -3.456878228130283e-10 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x))) < 6.019744398695255e-10

    1. Initial program 60.1

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

    2. Taylor expanded around inf 0.3

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

    3. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}}\]

    if 6.019744398695255e-10 < (- (/ (- 3) x) (/ (+ 1 (/ 3 x)) (* x x)))

    1. Initial program 0.5

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

    3. Applied add-exp-log0.5

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

Runtime

Time bar (total: 37.7s)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))