Average Error: 29.5 → 0.3
Time: 7.9m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\frac{-1}{x}\right) \cdot 3 + \left(\left(3 \cdot \frac{1}{3} + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x}\right))_* \le -1461976424.866403:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\frac{x + 1}{\sqrt[3]{x - 1}}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}\\ \mathbf{if}\;(\left(\frac{-1}{x}\right) \cdot 3 + \left(\left(3 \cdot \frac{1}{3} + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x}\right))_* \le 0.0004231168632593932:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{(\left(x + 1\right) \cdot \left((x \cdot x + x)_*\right) + \left(x + 1\right))_*}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (fma (/ -1 x) 3 (* (/ -1 (* x x)) (+ (/ 3 x) (* 1/3 3)))) < -1461976424.866403

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied prod-diff0.0

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

    8. Applied simplify0.0

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

    9. Applied simplify0.0

      \[\leadsto \left(\frac{x}{1 + x} - \frac{\frac{1 + x}{\sqrt[3]{x - 1}}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}\right) + \color{blue}{0}\]

    if -1461976424.866403 < (fma (/ -1 x) 3 (* (/ -1 (* x x)) (+ (/ 3 x) (* 1/3 3)))) < 0.0004231168632593932

    1. Initial program 58.6

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

    2. Taylor expanded around inf 0.8

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

    3. Applied simplify0.5

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

    if 0.0004231168632593932 < (fma (/ -1 x) 3 (* (/ -1 (* x x)) (+ (/ 3 x) (* 1/3 3))))

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied add-cube-cbrt0.1

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

    6. Applied prod-diff0.1

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

    7. Applied simplify0.1

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

    8. Applied simplify0.1

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

  4. Applied simplify0.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;(\left(\frac{-1}{x}\right) \cdot 3 + \left(\left(3 \cdot \frac{1}{3} + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x}\right))_* \le -1461976424.866403:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\frac{x + 1}{\sqrt[3]{x - 1}}}{\sqrt[3]{x - 1} \cdot \sqrt[3]{x - 1}}\\ \mathbf{if}\;(\left(\frac{-1}{x}\right) \cdot 3 + \left(\left(3 \cdot \frac{1}{3} + \frac{3}{x}\right) \cdot \frac{-1}{x \cdot x}\right))_* \le 0.0004231168632593932:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{(\left(x + 1\right) \cdot \left((x \cdot x + x)_*\right) + \left(x + 1\right))_*}{(\left(x \cdot x\right) \cdot x + \left(-1\right))_*}\\ \end{array}}\]

Runtime

Time bar (total: 7.9m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))