Average Error: 33.3 → 27.3
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -35445750.516213864:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right) + (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(\ell \cdot U\right) + \left(2 \cdot \ell\right))_*\right))_* \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right)}\\ \mathbf{elif}\;n \le 8.557110805693902 \cdot 10^{-150}:\\ \;\;\;\;\left|\sqrt{(\left((\left(\frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\ell \cdot \left(-2 \cdot U\right)}{\frac{Om}{n}}\right) + \left(\left(2 \cdot U\right) \cdot \left(t \cdot n\right)\right))_*}\right|\\ \mathbf{elif}\;n \le 3.614485940597305 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\sqrt{(\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot U\right)\right)\right) + \left(\sqrt[3]{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)} \cdot \sqrt[3]{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}\right) \cdot \sqrt[3]{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}}} \cdot \sqrt{\sqrt{(\left((\left(\frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot U\right)\right)\right) + \left(\left(t \cdot 2\right) \cdot \left(U \cdot n\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right) + (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(\ell \cdot U\right) + \left(2 \cdot \ell\right))_*\right))_* \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right)}\\ \end{array}\]

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 n < -35445750.516213864 or 3.614485940597305e+42 < n

    1. Initial program 31.7

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

    2. Initial simplification34.0

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

    4. Applied sub-neg34.0

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

    5. Applied distribute-rgt-in34.0

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

    6. Simplified32.0

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

    7. Taylor expanded around -inf 31.9

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

    8. Simplified29.4

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

    if -35445750.516213864 < n < 8.557110805693902e-150

    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 simplification32.2

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

    4. Applied sub-neg32.2

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

    5. Applied distribute-rgt-in32.2

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

    6. Simplified29.3

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

    8. Applied associate-*l*25.8

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

    10. Applied add-sqr-sqrt25.8

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

    11. Applied rem-sqrt-square25.8

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

    12. Simplified28.0

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

    if 8.557110805693902e-150 < n < 3.614485940597305e+42

    1. Initial program 31.8

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

    2. Initial simplification27.5

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

    4. Applied sub-neg27.5

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

    5. Applied distribute-rgt-in27.5

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

    6. Simplified22.8

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

    8. Applied associate-*l*20.6

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

    10. Applied add-cube-cbrt20.8

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

    12. Applied add-sqr-sqrt20.8

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

    13. Applied sqrt-prod21.0

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

    14. Simplified20.9

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

  4. Final simplification27.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -35445750.516213864:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right) + (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(\ell \cdot U\right) + \left(2 \cdot \ell\right))_*\right))_* \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right)}\\ \mathbf{elif}\;n \le 8.557110805693902 \cdot 10^{-150}:\\ \;\;\;\;\left|\sqrt{(\left((\left(\frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\ell \cdot \left(-2 \cdot U\right)}{\frac{Om}{n}}\right) + \left(\left(2 \cdot U\right) \cdot \left(t \cdot n\right)\right))_*}\right|\\ \mathbf{elif}\;n \le 3.614485940597305 \cdot 10^{+42}:\\ \;\;\;\;\sqrt{\sqrt{(\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot U\right)\right)\right) + \left(\sqrt[3]{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)} \cdot \sqrt[3]{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}\right) \cdot \sqrt[3]{t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}}} \cdot \sqrt{\sqrt{(\left((\left(\frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(-2 \cdot U\right)\right)\right) + \left(\left(t \cdot 2\right) \cdot \left(U \cdot n\right)\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right) + (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(\ell \cdot U\right) + \left(2 \cdot \ell\right))_*\right))_* \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018278 +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*))))))