Average Error: 42.4 → 9.5
Time: 1.3m
Precision: 64
Internal Precision: 1408
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.489564848441447 \cdot 10^{+80}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{x \cdot x} \cdot \frac{1}{\sqrt{2}} - (\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x}\right) + \left((\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x}\right) + \left(t \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{if}\;t \le 3.3413484077652453 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\frac{\sqrt{1}}{t}}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{t \cdot t}{\frac{x}{4}}\right))_*}}\\ \mathbf{if}\;t \le 8.33102543221026 \cdot 10^{-227}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{2}{x}\right) + \left(t \cdot \sqrt{2}\right))_* + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)}\\ \mathbf{if}\;t \le 1.598402083248483 \cdot 10^{+41}:\\ \;\;\;\;\left(\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}} \cdot \sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}}\right) \cdot \sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{t}{x} \cdot \left(t \cdot 4\right)\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{(\left(\frac{t}{\sqrt{2}}\right) \cdot \left(\frac{2}{x}\right) + \left(t \cdot \sqrt{2}\right))_* + \frac{t}{x \cdot x} \cdot \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -3.489564848441447e+80

    1. Initial program 48.0

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

    2. Applied simplify48.0

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

    3. Taylor expanded around -inf 2.9

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

    4. Applied simplify2.9

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

    if -3.489564848441447e+80 < t < 3.3413484077652453e-269

    1. Initial program 39.6

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

    2. Applied simplify39.6

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

    3. Taylor expanded around inf 19.1

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

    4. Applied simplify15.2

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

    6. Applied add-sqr-sqrt15.4

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

    7. Applied associate-*r*15.3

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

    9. Applied *-un-lft-identity15.3

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

    10. Applied sqrt-prod15.3

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

    11. Applied associate-/r*15.3

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

    12. Applied simplify15.3

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

    if 3.3413484077652453e-269 < t < 8.33102543221026e-227 or 1.598402083248483e+41 < t

    1. Initial program 45.1

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

    2. Applied simplify45.1

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

    3. Taylor expanded around inf 7.6

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

    4. Applied simplify7.6

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

    if 8.33102543221026e-227 < t < 1.598402083248483e+41

    1. Initial program 37.1

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

    2. Applied simplify37.0

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

    3. Taylor expanded around inf 15.2

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

    4. Applied simplify10.4

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

    6. Applied add-sqr-sqrt10.5

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

    7. Applied associate-*r*10.4

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

    9. Applied add-cube-cbrt10.6

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

    10. Applied simplify10.5

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

    11. Applied simplify10.4

      \[\leadsto \left(\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}} \cdot \sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\left(t \cdot 4\right) \cdot \frac{t}{x}\right))_*}}}\right) \cdot \color{blue}{\sqrt[3]{\frac{t \cdot \sqrt{2}}{\sqrt{(\left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) \cdot 2 + \left(\frac{t}{x} \cdot \left(t \cdot 4\right)\right))_*}}}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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)))))