Average Error: 33.8 → 30.4
Time: 51.4s
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -2.8040115762429476 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\frac{U}{\frac{Om}{\ell}} \cdot \frac{n}{\frac{Om}{\ell}} - \frac{n}{\frac{Om}{\ell}} \cdot \frac{U*}{\frac{Om}{\ell}}\right))_*\right)}\\ \mathbf{elif}\;U \le -6.079335643099636 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot 2} \cdot \sqrt{t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\frac{U}{\frac{Om}{\ell}} \cdot \frac{n}{\frac{Om}{\ell}} - \frac{n}{\frac{Om}{\ell}} \cdot \frac{U*}{\frac{Om}{\ell}}\right))_*}\\ \mathbf{elif}\;U \le 5.10091641267494 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) + \ell \cdot 2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)} \cdot \left(\sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\right)\right))_*\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

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 4 regimes

  2. if U < -2.8040115762429476e-223

    1. Initial program 32.5

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

    3. Applied *-un-lft-identity32.5

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

    4. Applied associate-*r*32.5

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

    5. Simplified29.1

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

    6. Taylor expanded around inf 37.5

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

    7. Simplified28.4

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

    if -2.8040115762429476e-223 < U < -6.079335643099636e-294

    1. Initial program 42.5

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

    3. Applied *-un-lft-identity42.5

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

    4. Applied associate-*r*42.5

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

    5. Simplified40.9

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

    6. Taylor expanded around inf 47.1

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

    7. Simplified38.8

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

    9. Applied sqrt-prod48.0

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

    10. Simplified48.0

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

    if -6.079335643099636e-294 < U < 5.10091641267494e-195

    1. Initial program 41.6

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

    3. Applied *-un-lft-identity41.6

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

    4. Applied associate-*r*41.6

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

    5. Simplified39.9

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

    7. Applied pow139.9

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

    8. Applied pow139.9

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

    9. Applied pow-prod-down39.9

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

    10. Simplified33.5

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

    if 5.10091641267494e-195 < U

    1. Initial program 30.4

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

    3. Applied *-un-lft-identity30.4

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

    4. Applied associate-*r*30.4

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

    5. Simplified27.6

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

    7. Applied add-cube-cbrt27.6

      \[\leadsto \sqrt{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \color{blue}{\left(\left(\sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\right) \cdot \sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\right)})_*\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification30.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.8040115762429476 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\frac{U}{\frac{Om}{\ell}} \cdot \frac{n}{\frac{Om}{\ell}} - \frac{n}{\frac{Om}{\ell}} \cdot \frac{U*}{\frac{Om}{\ell}}\right))_*\right)}\\ \mathbf{elif}\;U \le -6.079335643099636 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot 2} \cdot \sqrt{t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\frac{U}{\frac{Om}{\ell}} \cdot \frac{n}{\frac{Om}{\ell}} - \frac{n}{\frac{Om}{\ell}} \cdot \frac{U*}{\frac{Om}{\ell}}\right))_*}\\ \mathbf{elif}\;U \le 5.10091641267494 \cdot 10^{-195}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) + \ell \cdot 2\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)} \cdot \left(\sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)} \cdot \sqrt[3]{\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}\right)\right))_*\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +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*))))))