Average Error: 32.9 → 26.9
Time: 58.7s
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;Om \le 10950880976536786.0:\\ \;\;\;\;{\left(\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(\sqrt[3]{\left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right) \cdot n} \cdot \left(\sqrt[3]{\left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right) \cdot n} \cdot \sqrt[3]{\left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right) \cdot n}\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;Om \le 1.134410977021849 \cdot 10^{+237}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(U \cdot -2\right) \cdot \left(\ell \cdot n\right)}{Om}\right) + \left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right))_*}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)}}\\ \end{array}\]

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 Om < 10950880976536786.0

    1. Initial program 34.1

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

    2. Initial simplification33.1

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

    4. Applied sub-neg33.1

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

    5. Applied distribute-rgt-in33.1

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

    6. Simplified29.8

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

    8. Applied associate-*l*27.7

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

    10. Applied pow1/227.7

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

    12. Applied add-cube-cbrt27.8

      \[\leadsto {\left(t \cdot \left(2 \cdot \left(U \cdot n\right)\right) + \color{blue}{\left(\left(\sqrt[3]{n \cdot \left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{n \cdot \left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right)}\right) \cdot \sqrt[3]{n \cdot \left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right)}\right)} \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right)}^{\frac{1}{2}}\]

    if 10950880976536786.0 < Om < 1.134410977021849e+237

    1. Initial program 30.1

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

    2. Initial simplification28.1

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

    4. Applied sub-neg28.1

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

    5. Applied distribute-rgt-in28.1

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

    6. Simplified26.2

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

    8. Applied add-sqr-sqrt26.2

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

    9. Applied rem-sqrt-square26.2

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

    10. Simplified24.2

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

    if 1.134410977021849e+237 < Om

    1. Initial program 30.6

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

    2. Initial simplification26.1

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

    4. Applied add-sqr-sqrt26.3

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

  4. Final simplification26.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;Om \le 10950880976536786.0:\\ \;\;\;\;{\left(\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(\sqrt[3]{\left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right) \cdot n} \cdot \left(\sqrt[3]{\left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right) \cdot n} \cdot \sqrt[3]{\left(\left(U \cdot -2\right) \cdot \frac{\ell}{Om}\right) \cdot n}\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;Om \le 1.134410977021849 \cdot 10^{+237}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(U \cdot -2\right) \cdot \left(\ell \cdot n\right)}{Om}\right) + \left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right))_*}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)}}\\ \end{array}\]

Runtime

Time bar (total: 58.7s)Debug logProfile

herbie shell --seed 2018248 +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*))))))