Average Error: 29.2 → 0.2
Time: 3.0s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 8.51670179 \cdot 10^{-10}:\\ \;\;\;\;\frac{-\left(\frac{1}{x} + 3\right)}{x} - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))) < 8.51670179e-10

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

    5. Applied associate-*l/0.2

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

    6. Simplified0.2

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

    if 8.51670179e-10 < (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))

    1. Initial program 0.2

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

    3. Applied flip3--0.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{x + 1} - \frac{x + 1}{x - 1} \le 8.51670179 \cdot 10^{-10}:\\ \;\;\;\;\frac{-\left(\frac{1}{x} + 3\right)}{x} - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{x + \left(x + 1\right) \cdot \frac{x + 1}{x - 1}}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))