Average Error: 29.2 → 0.5
Time: 31.9s
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(1 + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x} + \left(-\frac{3}{x}\right) \le -9.330644946523259 \cdot 10^{-07}:\\ \;\;\;\;x \cdot \left(3 + x\right) + 1\\ \mathbf{if}\;\left(1 + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x} + \left(-\frac{3}{x}\right) \le 3.1549197993709034 \cdot 10^{-07}:\\ \;\;\;\;\left(1 + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x} + \left(-\frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (+ (* (+ 1 (/ 3 x)) (/ (- 1) (* x x))) (- (/ 3 x))) < -9.330644946523259e-07

    1. Initial program 0.2

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

    2. Taylor expanded around 0 1.7

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

    3. Applied simplify1.7

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

    if -9.330644946523259e-07 < (+ (* (+ 1 (/ 3 x)) (/ (- 1) (* x x))) (- (/ 3 x))) < 3.1549197993709034e-07

    1. Initial program 59.5

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

    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 3.1549197993709034e-07 < (+ (* (+ 1 (/ 3 x)) (/ (- 1) (* x x))) (- (/ 3 x)))

    1. Initial program 0.2

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

    3. Applied flip--0.2

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

Runtime

Time bar (total: 31.9s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))