Average Error: 42.3 → 10.5
Time: 38.2s
Precision: 64
Internal Precision: 128
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.450286884866024 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{x}\right) \cdot \left(\frac{t}{2 \cdot \sqrt{2}}\right) + \left(-(t \cdot \left(\sqrt{2}\right) + \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right))_*\right))_*}\\ \mathbf{elif}\;t \le -2.91378921224821 \cdot 10^{-160}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\left|\sqrt{(\left(\frac{t \cdot t}{x}\right) \cdot 4 + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right))_*}\right|} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\\ \mathbf{elif}\;t \le 9.911555569719438 \cdot 10^{-143}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left|(\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{2}{x}\right) + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right))_*\right|}\\ \mathbf{elif}\;t \le 5.5760203727373115 \cdot 10^{+78}:\\ \;\;\;\;\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\left|\sqrt{(\left(t \cdot t\right) \cdot \left(\frac{4}{x}\right) + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right))_*}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{x \cdot \sqrt{2}}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_* + \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right) \cdot \frac{\frac{2}{x}}{x}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 5 regimes

  2. if t < -1.450286884866024e+69

    1. Initial program 46.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{(\left(\frac{\frac{2}{x}}{x}\right) \cdot \left(\frac{t}{2 \cdot \sqrt{2}}\right) + \left(-(t \cdot \left(\sqrt{2}\right) + \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{\frac{2}{x}}{x} + \frac{2}{x}\right)\right))_*\right))_*}}\]

    if -1.450286884866024e+69 < t < -2.91378921224821e-160

    1. Initial program 27.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 9.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified9.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt9.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*} \cdot \sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}}}}\]

    6. Applied rem-sqrt-square9.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left|\sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}\right|}}\]

    7. Simplified4.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left|\color{blue}{\sqrt{(\left(t \cdot t\right) \cdot \left(\frac{4}{x}\right) + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right))_*}}\right|}\]

    8. Taylor expanded around -inf 4.7

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\left|\sqrt{(\left({t}^{2}\right) \cdot \left(\frac{4}{x}\right) + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left({t}^{2}\right))_*\right))_*}\right|}}\]

    9. Simplified4.7

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\left|\sqrt{(\left(\frac{t \cdot t}{x}\right) \cdot 4 + \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_* \cdot 2\right))_*}\right|} \cdot t}\]
    10. Using strategy rm

    11. Applied *-un-lft-identity4.7

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{1 \cdot \left|\sqrt{(\left(\frac{t \cdot t}{x}\right) \cdot 4 + \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_* \cdot 2\right))_*}\right|}} \cdot t\]

    12. Applied add-cube-cbrt4.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}{1 \cdot \left|\sqrt{(\left(\frac{t \cdot t}{x}\right) \cdot 4 + \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_* \cdot 2\right))_*}\right|} \cdot t\]

    13. Applied times-frac4.7

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{1} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\left|\sqrt{(\left(\frac{t \cdot t}{x}\right) \cdot 4 + \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_* \cdot 2\right))_*}\right|}\right)} \cdot t\]

    14. Simplified4.7

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\left|\sqrt{(\left(\frac{t \cdot t}{x}\right) \cdot 4 + \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_* \cdot 2\right))_*}\right|}\right) \cdot t\]

    if -2.91378921224821e-160 < t < 9.911555569719438e-143

    1. Initial program 59.4

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 32.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified32.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt32.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*} \cdot \sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}}}}\]

    6. Applied rem-sqrt-square32.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left|\sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}\right|}}\]

    7. Simplified30.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left|\color{blue}{\sqrt{(\left(t \cdot t\right) \cdot \left(\frac{4}{x}\right) + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right))_*}}\right|}\]

    8. Taylor expanded around -inf 30.2

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\left|\sqrt{(\left({t}^{2}\right) \cdot \left(\frac{4}{x}\right) + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left({t}^{2}\right))_*\right))_*}\right|}}\]

    9. Simplified30.2

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\left|\sqrt{(\left(\frac{t \cdot t}{x}\right) \cdot 4 + \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_* \cdot 2\right))_*}\right|} \cdot t}\]

    10. Taylor expanded around inf 36.0

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

    11. Simplified36.0

      \[\leadsto \frac{\sqrt{2}}{\left|\color{blue}{(\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{2}{x}\right) + \left(t \cdot \sqrt{2} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right))_*}\right|} \cdot t\]

    if 9.911555569719438e-143 < t < 5.5760203727373115e+78

    1. Initial program 26.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 10.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified10.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}}}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt10.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*} \cdot \sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}}}}\]

    6. Applied rem-sqrt-square10.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left|\sqrt{(\left(\frac{\ell \cdot \ell}{x}\right) \cdot 2 + \left(\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)\right))_*}\right|}}\]

    7. Simplified5.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\left|\color{blue}{\sqrt{(\left(t \cdot t\right) \cdot \left(\frac{4}{x}\right) + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right))_*}}\right|}\]
    8. Using strategy rm

    9. Applied div-inv5.6

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\left|\sqrt{(\left(t \cdot t\right) \cdot \left(\frac{4}{x}\right) + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right))_*}\right|}}\]

    if 5.5760203727373115e+78 < t

    1. Initial program 48.6

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 3.4

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

    3. Simplified3.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{(\left(\frac{t}{\sqrt{2} \cdot x}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_* + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right)}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.450286884866024 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{\frac{2}{x}}{x}\right) \cdot \left(\frac{t}{2 \cdot \sqrt{2}}\right) + \left(-(t \cdot \left(\sqrt{2}\right) + \left(\frac{t}{\sqrt{2}} \cdot \left(\frac{2}{x} + \frac{\frac{2}{x}}{x}\right)\right))_*\right))_*}\\ \mathbf{elif}\;t \le -2.91378921224821 \cdot 10^{-160}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\left|\sqrt{(\left(\frac{t \cdot t}{x}\right) \cdot 4 + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right))_*}\right|} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)\\ \mathbf{elif}\;t \le 9.911555569719438 \cdot 10^{-143}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left|(\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{2}{x}\right) + \left(\sqrt{2} \cdot t - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right))_*\right|}\\ \mathbf{elif}\;t \le 5.5760203727373115 \cdot 10^{+78}:\\ \;\;\;\;\left(\sqrt{2} \cdot t\right) \cdot \frac{1}{\left|\sqrt{(\left(t \cdot t\right) \cdot \left(\frac{4}{x}\right) + \left(2 \cdot (\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right))_*}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{x \cdot \sqrt{2}}\right) \cdot 2 + \left(\sqrt{2} \cdot t\right))_* + \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right) \cdot \frac{\frac{2}{x}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))