Average Error: 29.4 → 0.1
Time: 59.6s
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(1 + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x} + \left(-\frac{3}{x}\right) \le -5.860905052806863 \cdot 10^{-07}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}\\ \mathbf{if}\;\left(1 + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x} + \left(-\frac{3}{x}\right) \le 3.882718893686457 \cdot 10^{-05}:\\ \;\;\;\;\left(1 + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x} + \left(-\frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}} \cdot \sqrt{\frac{x}{x + 1} - \frac{x + 1}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (+ (* (+ 1 (/ 3 x)) (/ (- 1) (* x x))) (- (/ 3 x))) < -5.860905052806863e-07

    1. Initial program 0.2

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

    3. Applied frac-sub0.2

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

    if -5.860905052806863e-07 < (+ (* (+ 1 (/ 3 x)) (/ (- 1) (* x x))) (- (/ 3 x))) < 3.882718893686457e-05

    1. Initial program 59.4

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

    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

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

    if 3.882718893686457e-05 < (+ (* (+ 1 (/ 3 x)) (/ (- 1) (* x x))) (- (/ 3 x)))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.2

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

Runtime

Time bar (total: 59.6s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))