Average Error: 29.4 → 0.1
Time: 7.8m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\frac{\frac{1}{3}}{x}\right) \cdot \left(\frac{-3}{x}\right) + \left(\frac{-3}{x}\right))_* + \frac{\frac{-3}{x}}{x \cdot x} \le -3.954960597368782 \cdot 10^{-05}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{(\left(\frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*}\\ \mathbf{if}\;(\left(\frac{\frac{1}{3}}{x}\right) \cdot \left(\frac{-3}{x}\right) + \left(\frac{-3}{x}\right))_* + \frac{\frac{-3}{x}}{x \cdot x} \le 5.347362044394717 \cdot 10^{-08}:\\ \;\;\;\;(\left(\frac{\frac{1}{3}}{x}\right) \cdot \left(\frac{-3}{x}\right) + \left(\frac{-3}{x}\right))_* + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;(x \cdot \left(\frac{1}{x + 1}\right) + \left(-\frac{x + 1}{x - 1}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (+ (fma (/ 1/3 x) (/ -3 x) (/ -3 x)) (/ (/ -3 x) (* x x))) < -3.954960597368782e-05

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Applied simplify0.1

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

    if -3.954960597368782e-05 < (+ (fma (/ 1/3 x) (/ -3 x) (/ -3 x)) (/ (/ -3 x) (* x x))) < 5.347362044394717e-08

    1. Initial program 59.7

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

    3. Applied add-cbrt-cube59.7

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

    4. Applied simplify59.7

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

    5. Taylor expanded around inf 62.9

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

    6. Applied simplify0.0

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

    if 5.347362044394717e-08 < (+ (fma (/ 1/3 x) (/ -3 x) (/ -3 x)) (/ (/ -3 x) (* x x)))

    1. Initial program 0.2

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

    3. Applied div-inv0.2

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

    4. Applied fma-neg0.2

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

Runtime

Time bar (total: 7.8m)Debug logProfile

herbie shell --seed 2018201 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))