Average Error: 29.2 → 0.1
Time: 9.4m
Precision: 64
Internal Precision: 1344
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(-\frac{9}{x}\right) - \frac{\frac{48}{x}}{x \cdot x}\right) - \frac{12}{x \cdot x}}{(\left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) \cdot \left(\frac{1 + x}{x - 1}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*} \le -1.7505084052522833 \cdot 10^{-08}:\\ \;\;\;\;\frac{(\left(\frac{\frac{1 + x}{x - 1}}{\left(x - 1\right) \cdot \left(x - 1\right)}\right) \cdot \left(\left(1 + x\right) \cdot \left(-\left(1 + x\right)\right)\right) + \left(\frac{{x}^{3}}{{\left(1 + x\right)}^{3}}\right))_* + \left(\frac{{\left(1 + x\right)}^{3}}{{\left(x - 1\right)}^{3}} - \frac{{\left(1 + x\right)}^{3}}{{\left(x - 1\right)}^{3}}\right)}{(\left(\frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*}\\ \mathbf{if}\;\frac{\left(\left(-\frac{9}{x}\right) - \frac{\frac{48}{x}}{x \cdot x}\right) - \frac{12}{x \cdot x}}{(\left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) \cdot \left(\frac{1 + x}{x - 1}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*} \le 1.9919282774550476 \cdot 10^{-05}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-1}{x - 1}\right) \cdot \left(1 + x\right) + \left(\frac{x}{1 + x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (- (- (/ 9 x)) (/ (/ 48 x) (* x x))) (/ 12 (* x x))) (fma (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x))) (/ (+ 1 x) (- x 1)) (* (/ x (+ 1 x)) (/ x (+ 1 x))))) < -1.7505084052522833e-08

    1. Initial program 0.3

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

    3. Applied flip3--0.3

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

    4. Applied simplify0.3

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

    6. Applied *-un-lft-identity0.3

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

    7. Applied add-cube-cbrt0.4

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

    8. Applied times-frac0.4

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

    9. Applied unpow-prod-down0.4

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

    10. Applied div-inv0.4

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

    11. Applied unpow-prod-down0.4

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

    12. Applied prod-diff0.4

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

    13. Applied simplify0.3

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

    14. Applied simplify0.2

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

    if -1.7505084052522833e-08 < (/ (- (- (- (/ 9 x)) (/ (/ 48 x) (* x x))) (/ 12 (* x x))) (fma (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x))) (/ (+ 1 x) (- x 1)) (* (/ x (+ 1 x)) (/ x (+ 1 x))))) < 1.9919282774550476e-05

    1. Initial program 59.8

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

    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 1.9919282774550476e-05 < (/ (- (- (- (/ 9 x)) (/ (/ 48 x) (* x x))) (/ 12 (* x x))) (fma (+ (/ (+ 1 x) (- x 1)) (/ x (+ 1 x))) (/ (+ 1 x) (- x 1)) (* (/ x (+ 1 x)) (/ x (+ 1 x)))))

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Applied associate-/r/0.1

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

    5. Applied add-sqr-sqrt61.0

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

    6. Applied prod-diff61.0

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

    7. Applied simplify0.1

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

    8. Applied simplify0.1

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

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\left(\left(-\frac{9}{x}\right) - \frac{\frac{48}{x}}{x \cdot x}\right) - \frac{12}{x \cdot x}}{(\left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) \cdot \left(\frac{1 + x}{x - 1}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*} \le -1.7505084052522833 \cdot 10^{-08}:\\ \;\;\;\;\frac{(\left(\frac{\frac{1 + x}{x - 1}}{\left(x - 1\right) \cdot \left(x - 1\right)}\right) \cdot \left(\left(1 + x\right) \cdot \left(-\left(1 + x\right)\right)\right) + \left(\frac{{x}^{3}}{{\left(1 + x\right)}^{3}}\right))_* + \left(\frac{{\left(1 + x\right)}^{3}}{{\left(x - 1\right)}^{3}} - \frac{{\left(1 + x\right)}^{3}}{{\left(x - 1\right)}^{3}}\right)}{(\left(\frac{1 + x}{x - 1}\right) \cdot \left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*}\\ \mathbf{if}\;\frac{\left(\left(-\frac{9}{x}\right) - \frac{\frac{48}{x}}{x \cdot x}\right) - \frac{12}{x \cdot x}}{(\left(\frac{x}{1 + x} + \frac{1 + x}{x - 1}\right) \cdot \left(\frac{1 + x}{x - 1}\right) + \left(\frac{x}{1 + x} \cdot \frac{x}{1 + x}\right))_*} \le 1.9919282774550476 \cdot 10^{-05}:\\ \;\;\;\;(\left(1 + \frac{3}{x}\right) \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\frac{-3}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{-1}{x - 1}\right) \cdot \left(1 + x\right) + \left(\frac{x}{1 + x}\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 9.4m)Debug logProfile

herbie shell --seed 2018207 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))