Average Error: 33.5 → 32.4
Time: 3.0m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -2.190105908957428 \cdot 10^{-247}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}\\ \mathbf{elif}\;U \le 1.393467666117896 \cdot 10^{-251}:\\ \;\;\;\;{\left(\left(\left((-2 \cdot \left(\frac{\ell \cdot \ell}{Om}\right) + t)_* - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right) \cdot n\right) \cdot \left(2 \cdot U\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le 1.558504887377046 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot U\right) \cdot -2\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left((-2 \cdot \left(\frac{\ell \cdot \ell}{Om}\right) + t)_* - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right) \cdot n\right) \cdot \left(2 \cdot U\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

  1. Split input into 3 regimes

  2. if U < -2.190105908957428e-247

    1. Initial program 32.1

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

    2. Taylor expanded around -inf 36.0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\frac{n \cdot {\ell}^{2}}{{Om}^{2}}} \cdot \left(U - U*\right)\right)}\]

    3. Simplified31.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}} \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity31.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}\]

    6. Applied times-frac28.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)}\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}\]

    7. Simplified28.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\color{blue}{\ell} \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt29.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt29.3

      \[\leadsto \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}}} \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}\]

    if -2.190105908957428e-247 < U < 1.393467666117896e-251 or 1.558504887377046e-131 < U

    1. Initial program 33.5

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied pow133.5

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}^{1}}}\]

    4. Applied pow133.5

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot \color{blue}{{U}^{1}}\right) \cdot {\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}^{1}}\]

    5. Applied pow133.5

      \[\leadsto \sqrt{\left(\left(2 \cdot \color{blue}{{n}^{1}}\right) \cdot {U}^{1}\right) \cdot {\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}^{1}}\]

    6. Applied pow133.5

      \[\leadsto \sqrt{\left(\left(\color{blue}{{2}^{1}} \cdot {n}^{1}\right) \cdot {U}^{1}\right) \cdot {\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}^{1}}\]

    7. Applied pow-prod-down33.5

      \[\leadsto \sqrt{\left(\color{blue}{{\left(2 \cdot n\right)}^{1}} \cdot {U}^{1}\right) \cdot {\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}^{1}}\]

    8. Applied pow-prod-down33.5

      \[\leadsto \sqrt{\color{blue}{{\left(\left(2 \cdot n\right) \cdot U\right)}^{1}} \cdot {\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}^{1}}\]

    9. Applied pow-prod-down33.5

      \[\leadsto \sqrt{\color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}^{1}}}\]

    10. Applied sqrt-pow133.5

      \[\leadsto \color{blue}{{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}^{\left(\frac{1}{2}\right)}}\]

    11. Simplified33.7

      \[\leadsto {\color{blue}{\left(\left(\left((-2 \cdot \left(\frac{\ell \cdot \ell}{Om}\right) + t)_* - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right)\right)}}^{\left(\frac{1}{2}\right)}\]

    if 1.393467666117896e-251 < U < 1.558504887377046e-131

    1. Initial program 38.4

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied sub-neg38.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]

    4. Applied distribute-lft-in38.4

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\]

    5. Simplified38.1

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \color{blue}{\left(\left(-2 \cdot \left(U \cdot n\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification32.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.190105908957428 \cdot 10^{-247}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}}\\ \mathbf{elif}\;U \le 1.393467666117896 \cdot 10^{-251}:\\ \;\;\;\;{\left(\left(\left((-2 \cdot \left(\frac{\ell \cdot \ell}{Om}\right) + t)_* - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right) \cdot n\right) \cdot \left(2 \cdot U\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le 1.558504887377046 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(n \cdot U\right) \cdot -2\right)\right) + \left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left((-2 \cdot \left(\frac{\ell \cdot \ell}{Om}\right) + t)_* - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right)\right) \cdot n\right) \cdot \left(2 \cdot U\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))