Average Error: 29.0 → 0.3
Time: 1.6m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \le 4.178112254527817 \cdot 10^{-10}:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left((x \cdot x + \left(1 - x\right))_*\right) \cdot \left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))) < 4.178112254527817e-10

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.6

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

    3. Applied simplify0.3

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

    if 4.178112254527817e-10 < (- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

    4. Applied flip3-+0.2

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

    5. Applied associate-/r/0.2

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

    6. Applied prod-diff0.2

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

    7. Applied simplify0.2

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

    8. Applied simplify0.2

      \[\leadsto (\left((x \cdot x + \left(1 - x\right))_*\right) \cdot \left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) + \left(-\frac{x + 1}{x - 1}\right))_* + \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \le 4.178112254527817 \cdot 10^{-10}:\\ \;\;\;\;(\left(\frac{3}{x} + 1\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(-\frac{3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left((x \cdot x + \left(1 - x\right))_*\right) \cdot \left(\frac{x}{(\left(x \cdot x\right) \cdot x + 1)_*}\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018170 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))