Average Error: 29.3 → 0.1
Time: 2.0m
Precision: 64
Internal Precision: 128
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -19673.352462380622:\\ \;\;\;\;\frac{(\left(\frac{1}{x \cdot x}\right) \cdot \left(\frac{-16}{x} + -5\right) + \left(\frac{-6}{x}\right))_*}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{elif}\;x \le 14178.207091281372:\\ \;\;\;\;\frac{\frac{(\left(x + -1\right) \cdot \left(\frac{x \cdot x}{1 + x}\right) + \left(\frac{1 + x}{x + -1} \cdot \left(\left(-1 - x\right) \cdot \left(1 + x\right)\right)\right))_*}{\left(x - 1\right) \cdot \left(1 + x\right)}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -19673.352462380622

    1. Initial program 59.3

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

    3. Applied flip--59.3

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

    5. Applied fma-neg59.3

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

    6. Taylor expanded around inf 0.4

      \[\leadsto \frac{\color{blue}{-\left(16 \cdot \frac{1}{{x}^{3}} + \left(5 \cdot \frac{1}{{x}^{2}} + 6 \cdot \frac{1}{x}\right)\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

    7. Simplified0.0

      \[\leadsto \frac{\color{blue}{(\left(\frac{1}{x \cdot x}\right) \cdot \left(-5 + \frac{-16}{x}\right) + \left(\frac{-6}{x}\right))_*}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

    if -19673.352462380622 < x < 14178.207091281372

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    5. Applied associate-*l/0.1

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

    6. Applied associate-*r/0.1

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

    7. Applied frac-sub0.1

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

    8. Simplified0.1

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

    if 14178.207091281372 < x

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -19673.352462380622:\\ \;\;\;\;\frac{(\left(\frac{1}{x \cdot x}\right) \cdot \left(\frac{-16}{x} + -5\right) + \left(\frac{-6}{x}\right))_*}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{elif}\;x \le 14178.207091281372:\\ \;\;\;\;\frac{\frac{(\left(x + -1\right) \cdot \left(\frac{x \cdot x}{1 + x}\right) + \left(\frac{1 + x}{x + -1} \cdot \left(\left(-1 - x\right) \cdot \left(1 + x\right)\right)\right))_*}{\left(x - 1\right) \cdot \left(1 + x\right)}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-1}{x \cdot x}\right) \cdot \left(\frac{3}{x}\right) + \left(\frac{-1}{x \cdot x} - \frac{3}{x}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))