Average Error: 29.5 → 0.6
Time: 4.7m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_* \le -1.6839866920797528 \cdot 10^{-11}:\\ \;\;\;\;(x \cdot \left(3 + x\right) + 1)_*\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_* \le 1.4542143759088224 \cdot 10^{-05}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < -1.6839866920797528e-11

    1. Initial program 0.5

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

    2. Taylor expanded around 0 2.3

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

    3. Applied simplify2.3

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

    if -1.6839866920797528e-11 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < 1.4542143759088224e-05

    1. Initial program 59.8

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

    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 1.4542143759088224e-05 < (fma (+ 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 flip--0.1

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

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed 2018195 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))