Average Error: 29.3 → 0.1
Time: 3.1m
Precision: 64
Internal Precision: 1408
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -9.751436196247666 \cdot 10^{-09}:\\ \;\;\;\;\frac{x}{1 + x} - \left(x \cdot x + \left(1 + x\right)\right) \cdot \frac{1 + x}{{x}^{3} - {1}^{3}}\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 2.3971953008053916 \cdot 10^{-05}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{x}{{1}^{3} + {x}^{3}}\right) \cdot \left(\left(1 - x\right) + x \cdot x\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) < -9.751436196247666e-09

    1. Initial program 0.3

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

    3. Applied flip3--0.3

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

    4. Applied associate-/r/0.3

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

    if -9.751436196247666e-09 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (/ (- 3) x)) < 2.3971953008053916e-05

    1. Initial program 59.7

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

    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 2.3971953008053916e-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 flip3-+0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied fma-neg0.1

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

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -9.751436196247666 \cdot 10^{-09}:\\ \;\;\;\;\frac{x}{1 + x} - \left(x \cdot x + \left(1 + x\right)\right) \cdot \frac{1 + x}{{x}^{3} - {1}^{3}}\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 2.3971953008053916 \cdot 10^{-05}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{x}{{1}^{3} + {x}^{3}}\right) \cdot \left(\left(1 - x\right) + x \cdot x\right) + \left(\frac{-\left(1 + x\right)}{x - 1}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))