Error

Derivation

1. Split input into 3 regimes

2. if t < -6.654487547109705e+20

   1. Initial program 42.4

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

   2. Taylor expanded around -inf 4.9

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

   3. Simplified4.9

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

   if -6.654487547109705e+20 < t < 4.889623337220024e+113

   1. Initial program 39.1

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

   2. Taylor expanded around inf 18.6

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

   3. Simplified14.4

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

   4. Taylor expanded around -inf 18.6

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

   5. Simplified14.4

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

   if 4.889623337220024e+113 < t

   1. Initial program 52.1

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

   2. Taylor expanded around inf 2.9

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

   3. Simplified2.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{(\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right) + \left((\left(\frac{-1}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x}\right) + \left(\sqrt{2} \cdot t\right))_*\right))_*}}\]
3. Recombined 3 regimes into one program.

4. Final simplification9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.654487547109705 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{t}{x \cdot x}\right) \cdot \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) + \left((\left(\frac{2}{x}\right) \cdot \left(\frac{-t}{\sqrt{2}}\right) + \left(\left(-t\right) \cdot \sqrt{2}\right))_*\right))_*}\\ \mathbf{elif}\;t \le 4.889623337220024 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{(2 \cdot \left((\left(\frac{\ell}{x}\right) \cdot \ell + \left(t \cdot t\right))_*\right) + \left(\left(4 \cdot t\right) \cdot \frac{t}{x}\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{(\left(\frac{2}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left((\left(\frac{-1}{\sqrt{2}}\right) \cdot \left(\frac{t}{x \cdot x}\right) + \left(\sqrt{2} \cdot t\right))_*\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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)))))

Details

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