Average Error: 29.2 → 0.5
Time: 1.2m
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -9.330644946523259 \cdot 10^{-07}:\\ \;\;\;\;(x \cdot \left(3 + x\right) + 1)_*\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 3.1549197993709034 \cdot 10^{-07}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{x}{x + 1}\right) \cdot \left(\frac{x}{x + 1}\right) + \left(-\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right))_*}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 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 < (fma (+ 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

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

    if 3.1549197993709034e-07 < (fma (+ 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

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

    5. Applied fma-neg0.2

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

Runtime

Time bar (total: 1.2m)Debug logProfile

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