Average Error: 29.2 → 0.8
Time: 9.2m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -3.8411611046864326 \cdot 10^{-10}:\\ \;\;\;\;(x \cdot \left(3 + x\right) + 1)_*\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 8.133924293901448 \cdot 10^{-10}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left({\left(\frac{x}{x \cdot x - 1}\right)}^{3}\right) \cdot \left({\left(x - 1\right)}^{3}\right) + \left({\left(\frac{1 + x}{x \cdot x - 1}\right)}^{3} \cdot \left(-{\left(1 + x\right)}^{3}\right)\right))_* + (\left(-{\left(1 + x\right)}^{3}\right) \cdot \left({\left(\frac{1 + x}{x \cdot x - 1}\right)}^{3}\right) + \left({\left(1 + x\right)}^{3} \cdot {\left(\frac{1 + x}{x \cdot x - 1}\right)}^{3}\right))_*}{(\left(\frac{1 + x}{x - 1}\right) \cdot \left(\frac{1 + x}{x - 1} + \frac{x}{1 + x}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < -3.8411611046864326e-10

    1. Initial program 0.4

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

    2. Taylor expanded around 0 2.4

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

    3. Applied simplify2.4

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

    if -3.8411611046864326e-10 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x))) < 8.133924293901448e-10

    1. Initial program 60.1

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

    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 8.133924293901448e-10 < (fma (+ 1 (/ 3 x)) (/ (- 1) (* x x)) (- (/ 3 x)))

    1. Initial program 0.5

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

    3. Applied flip3--0.5

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

    4. Applied simplify0.5

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

    6. Applied flip--0.5

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

    7. Applied associate-/r/0.5

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

    8. Applied unpow-prod-down0.5

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

    9. Applied flip-+0.5

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

    10. Applied associate-/r/0.5

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

    11. Applied unpow-prod-down0.5

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

    12. Applied prod-diff0.5

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

  4. Applied simplify0.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le -3.8411611046864326 \cdot 10^{-10}:\\ \;\;\;\;(x \cdot \left(3 + x\right) + 1)_*\\ \mathbf{if}\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_* \le 8.133924293901448 \cdot 10^{-10}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left({\left(\frac{x}{x \cdot x - 1}\right)}^{3}\right) \cdot \left({\left(x - 1\right)}^{3}\right) + \left({\left(\frac{1 + x}{x \cdot x - 1}\right)}^{3} \cdot \left(-{\left(1 + x\right)}^{3}\right)\right))_* + (\left(-{\left(1 + x\right)}^{3}\right) \cdot \left({\left(\frac{1 + x}{x \cdot x - 1}\right)}^{3}\right) + \left({\left(1 + x\right)}^{3} \cdot {\left(\frac{1 + x}{x \cdot x - 1}\right)}^{3}\right))_*}{(\left(\frac{1 + x}{x - 1}\right) \cdot \left(\frac{1 + x}{x - 1} + \frac{x}{1 + x}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*}\\ \end{array}}\]

Runtime

Time bar (total: 9.2m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))