Average Error: 33.7 → 23.5
Time: 3.7m
Precision: 64
Internal Precision: 384
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(U \cdot n\right) \le -2.415523017271143 \cdot 10^{-236}:\\ \;\;\;\;\sqrt{(\left((\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{U* - U}{\frac{Om}{\ell}}\right) + t)_*\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right) + \left(\left(\frac{2}{Om} \cdot \left(-\ell\right)\right) \cdot \left(\left(\ell \cdot U\right) \cdot \left(n \cdot 2\right)\right)\right))_* + \left(n \cdot 2\right) \cdot \left(U \cdot 0\right)}\\ \mathbf{if}\;2 \cdot \left(U \cdot n\right) \le 3.065555060109775 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left((\left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{n}{\frac{Om}{\ell}}\right) + t)_* - \frac{2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot n\right)} \cdot \sqrt{(\left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{n}{\frac{Om}{\ell}}\right) + t)_* - \frac{2 \cdot \ell}{\frac{Om}{\ell}}}\\ \end{array}\]

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 (* 2 (* U n)) < -2.415523017271143e-236

    1. Initial program 27.4

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

    3. Applied add-sqr-sqrt59.2

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

    4. Applied prod-diff59.2

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

    5. Applied distribute-rgt-in59.2

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

    6. Applied simplify23.7

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

    7. Applied simplify23.2

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \color{blue}{\left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}}\]

    8. Taylor expanded around 0 24.1

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \color{blue}{\frac{n \cdot \ell}{Om}}\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\]

    9. Applied simplify23.4

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

    11. Applied add-sqr-sqrt31.5

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

    12. Applied add-sqr-sqrt57.6

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

    13. Applied prod-diff57.6

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

    14. Applied distribute-lft-in57.6

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

    15. Applied simplify32.7

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

    16. Applied simplify24.3

      \[\leadsto \sqrt{(\left((\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{U* - U}{\frac{Om}{\ell}}\right) + t)_*\right) \cdot \left(n \cdot \left(2 \cdot U\right)\right) + \left(\left(\frac{2}{Om} \cdot \left(-\ell\right)\right) \cdot \left(\left(\ell \cdot U\right) \cdot \left(n \cdot 2\right)\right)\right))_* + \color{blue}{\left(n \cdot 2\right) \cdot \left(U \cdot 0\right)}}\]

    if -2.415523017271143e-236 < (* 2 (* U n)) < 3.065555060109775e-287

    1. Initial program 51.3

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

    3. Applied add-sqr-sqrt57.9

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

    4. Applied prod-diff57.9

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

    5. Applied distribute-rgt-in57.9

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

    6. Applied simplify50.3

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

    7. Applied simplify49.6

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \color{blue}{\left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}}\]

    8. Taylor expanded around 0 49.7

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \color{blue}{\frac{n \cdot \ell}{Om}}\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\]

    9. Applied simplify49.7

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

    11. Applied associate-*l*35.1

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

    if 3.065555060109775e-287 < (* 2 (* U n))

    1. Initial program 28.7

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

    3. Applied add-sqr-sqrt30.8

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

    4. Applied prod-diff30.8

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

    5. Applied distribute-rgt-in30.8

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

    6. Applied simplify24.3

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

    7. Applied simplify23.5

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \color{blue}{\left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}}\]

    8. Taylor expanded around 0 24.8

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \color{blue}{\frac{n \cdot \ell}{Om}}\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\]

    9. Applied simplify23.7

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

    11. Applied sqrt-prod15.8

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

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +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*))))))