Average Error: 33.4 → 26.2
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le 2.2089170394906 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(2 \cdot \left(U \cdot n\right)\right) \cdot (\left(\frac{-n}{\frac{Om}{\ell}}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell \cdot -2}{\frac{Om}{\ell}}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U}\\ \end{array}\]

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 2 regimes

  2. if n < 2.2089170394906e-310

    1. Initial program 32.3

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

    3. Applied associate-/l*29.9

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

    5. Applied unpow229.9

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

    6. Applied associate-*r*29.2

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

    8. Applied associate-*l*29.2

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

    10. Applied sub-neg29.2

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

    11. Applied associate--l+29.2

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

    12. Applied distribute-lft-in29.2

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

    13. Applied distribute-lft-in29.2

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

    14. Simplified29.8

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

    if 2.2089170394906e-310 < n

    1. Initial program 34.6

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

    3. Applied associate-/l*32.1

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

    5. Applied unpow232.1

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

    6. Applied associate-*r*31.1

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

    8. Applied associate-*l*30.5

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

    10. Applied sqrt-prod22.5

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

  4. Final simplification26.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le 2.2089170394906 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(2 \cdot \left(U \cdot n\right)\right) \cdot (\left(\frac{-n}{\frac{Om}{\ell}}\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell \cdot -2}{\frac{Om}{\ell}}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +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*))))))