Average Error: 33.3 → 27.2
Time: 2.5m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -5.771197405462805 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{(\left(\frac{\ell}{\frac{Om}{\ell}}\right) \cdot -2 + t)_* \cdot \left(2 \cdot \left(n \cdot U\right)\right) + \left(U \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot -2}\\ \mathbf{elif}\;U \le 9.070015015746773 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(U* - U\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) + \left((\ell \cdot \left(\frac{-2}{\frac{Om}{\ell}}\right) + t)_* \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\frac{\ell}{\frac{Om}{\ell}}\right) \cdot -2 + t)_* \cdot \left(2 \cdot \left(n \cdot U\right)\right) + \left(U \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot -2}\\ \end{array}\]

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 2 regimes

  2. if U < -5.771197405462805e-28 or 9.070015015746773e-109 < U

    1. Initial program 27.9

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

    2. Initial simplification27.9

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

    4. Applied associate-/l*24.7

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

    6. Applied associate-*r*23.9

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

    8. Applied sub-neg23.9

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

    9. Applied distribute-lft-in23.9

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

    10. Simplified23.3

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

    if -5.771197405462805e-28 < U < 9.070015015746773e-109

    1. Initial program 38.7

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

    2. Initial simplification38.7

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

    4. Applied associate-/l*36.5

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

    6. Applied associate-*r*35.6

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

    8. Applied *-un-lft-identity35.6

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

    9. Applied associate-*r*35.6

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

    10. Simplified35.6

      \[\leadsto \sqrt{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot 1\right) \cdot \color{blue}{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(U* - U\right) + \left((\left(\frac{\ell}{\frac{Om}{\ell}}\right) \cdot -2 + t)_*\right))_*}}\]
    11. Using strategy rm

    12. Applied fma-udef35.6

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

    13. Applied distribute-rgt-in35.6

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

    14. Simplified31.0

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

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -5.771197405462805 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{(\left(\frac{\ell}{\frac{Om}{\ell}}\right) \cdot -2 + t)_* \cdot \left(2 \cdot \left(n \cdot U\right)\right) + \left(U \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot -2}\\ \mathbf{elif}\;U \le 9.070015015746773 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\left(U* - U\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) + \left((\ell \cdot \left(\frac{-2}{\frac{Om}{\ell}}\right) + t)_* \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\frac{\ell}{\frac{Om}{\ell}}\right) \cdot -2 + t)_* \cdot \left(2 \cdot \left(n \cdot U\right)\right) + \left(U \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot -2}\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018234 +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*))))))