Average Error: 33.4 → 24.4
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -2.1944020368849097 \cdot 10^{+144}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{n \cdot \ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\left(\left(U \cdot 2\right) \cdot \left(-n\right)\right) \cdot \frac{\ell}{Om}\right) + \left(\left(t \cdot 2\right) \cdot \left(n \cdot U\right)\right))_*}\right|\\ \mathbf{elif}\;U \le 1.8354039042807684 \cdot 10^{+33}:\\ \;\;\;\;{\left((\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\left(-U \cdot 2\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot t + \left(\left(-\frac{\ell}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot (\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}} \cdot \sqrt[3]{\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot t + \left(\left(-\frac{\ell}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot (\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot t + \left(\left(-\frac{\ell}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot (\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}}\\ \end{array}\]

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 U < -2.1944020368849097e+144

    1. Initial program 27.2

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

    2. Initial simplification25.1

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

    4. Applied sub-neg25.1

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

    5. Applied distribute-rgt-in25.1

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

    6. Simplified24.5

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

    8. Applied associate-*l*26.0

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

    10. Applied add-sqr-sqrt26.0

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

    11. Applied rem-sqrt-square26.0

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

    12. Simplified24.6

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

    if -2.1944020368849097e+144 < U < 1.8354039042807684e+33

    1. Initial program 35.1

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

    2. Initial simplification34.4

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

    4. Applied sub-neg34.4

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

    5. Applied distribute-rgt-in34.4

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

    6. Simplified30.8

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

    8. Applied associate-*l*28.3

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

    10. Applied pow1/228.3

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

    12. Applied pow128.3

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

    13. Applied pow-pow28.3

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

    14. Simplified24.3

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

    if 1.8354039042807684e+33 < U

    1. Initial program 28.6

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

    2. Initial simplification26.3

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

    4. Applied sub-neg26.3

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

    5. Applied distribute-rgt-in26.3

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

    6. Simplified24.3

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

    8. Applied add-cube-cbrt25.0

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

  4. Final simplification24.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.1944020368849097 \cdot 10^{+144}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{n \cdot \ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\left(\left(U \cdot 2\right) \cdot \left(-n\right)\right) \cdot \frac{\ell}{Om}\right) + \left(\left(t \cdot 2\right) \cdot \left(n \cdot U\right)\right))_*}\right|\\ \mathbf{elif}\;U \le 1.8354039042807684 \cdot 10^{+33}:\\ \;\;\;\;{\left((\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\left(-U \cdot 2\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot t + \left(\left(-\frac{\ell}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot (\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}} \cdot \sqrt[3]{\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot t + \left(\left(-\frac{\ell}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot (\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(n \cdot U\right) \cdot 2\right) \cdot t + \left(\left(-\frac{\ell}{Om}\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)\right) \cdot (\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018219 +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*))))))