Average Error: 33.0 → 28.4
Time: 59.2s
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;Om \le -6.890552229095882 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t + (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(U \cdot \ell\right) + \left(2 \cdot \ell\right))_*\right))_* \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U \cdot n\right) \cdot \left(-2\right)\right)\right)}\\ \mathbf{elif}\;Om \le -7.796636058379547 \cdot 10^{-162}:\\ \;\;\;\;\left|\sqrt[3]{(\left((\left(\frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{U \cdot -2}{\frac{Om}{\ell \cdot n}}\right) + \left(\left(t \cdot 2\right) \cdot \left(U \cdot n\right)\right))_*}\right| \cdot \sqrt{\sqrt[3]{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t + (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(\frac{U}{\frac{Om}{\ell \cdot n}} \cdot -2\right)}}\\ \mathbf{elif}\;Om \le 4.486731434938353 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t + (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(U \cdot \ell\right) + \left(2 \cdot \ell\right))_*\right))_* \cdot \left(\frac{U}{\frac{Om}{\ell \cdot n}} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U \cdot n\right) \cdot \left(-2\right)\right)\right) + \left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \end{array}\]

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 4 regimes

  2. if Om < -6.890552229095882e+16

    1. Initial program 29.8

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

    2. Initial simplification27.2

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

    4. Applied sub-neg27.2

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

    5. Applied distribute-rgt-in27.2

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

    6. Simplified25.9

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

    7. Taylor expanded around 0 28.4

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

    8. Simplified27.2

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

    if -6.890552229095882e+16 < Om < -7.796636058379547e-162

    1. Initial program 36.0

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

    2. Initial simplification36.8

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

    4. Applied sub-neg36.8

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

    5. Applied distribute-rgt-in36.8

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

    6. Simplified31.1

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

    7. Taylor expanded around 0 28.6

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

    9. Applied associate-/l*27.4

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

    11. Applied add-cube-cbrt27.7

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

    12. Applied sqrt-prod27.7

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

    13. Simplified27.8

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

    if -7.796636058379547e-162 < Om < 4.486731434938353e+119

    1. Initial program 36.1

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

    2. Initial simplification36.0

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

    4. Applied sub-neg36.0

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

    5. Applied distribute-rgt-in35.9

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

    6. Simplified31.9

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

    7. Taylor expanded around 0 32.7

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

    9. Applied associate-/l*30.0

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

    10. Taylor expanded around 0 28.7

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

    11. Simplified30.7

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

    if 4.486731434938353e+119 < Om

    1. Initial program 30.3

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

    2. Initial simplification27.3

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

    4. Applied sub-neg27.3

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

    5. Applied distribute-rgt-in27.3

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

    6. Simplified26.5

      \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(U \cdot n\right)\right) + \color{blue}{\left(\left(\left(n \cdot U\right) \cdot \left(-2\right)\right) \cdot \frac{\ell}{Om}\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification28.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;Om \le -6.890552229095882 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t + (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(U \cdot \ell\right) + \left(2 \cdot \ell\right))_*\right))_* \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U \cdot n\right) \cdot \left(-2\right)\right)\right)}\\ \mathbf{elif}\;Om \le -7.796636058379547 \cdot 10^{-162}:\\ \;\;\;\;\left|\sqrt[3]{(\left((\left(\frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{U \cdot -2}{\frac{Om}{\ell \cdot n}}\right) + \left(\left(t \cdot 2\right) \cdot \left(U \cdot n\right)\right))_*}\right| \cdot \sqrt{\sqrt[3]{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t + (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(\frac{U}{\frac{Om}{\ell \cdot n}} \cdot -2\right)}}\\ \mathbf{elif}\;Om \le 4.486731434938353 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot t + (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(U \cdot \ell\right) + \left(2 \cdot \ell\right))_*\right))_* \cdot \left(\frac{U}{\frac{Om}{\ell \cdot n}} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(\frac{\ell}{Om} \cdot \left(\left(U \cdot n\right) \cdot \left(-2\right)\right)\right) + \left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \end{array}\]

Runtime

Time bar (total: 59.2s)Debug logProfile

herbie shell --seed 2018225 +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*))))))