Average Error: 33.0 → 30.9
Time: 5.7m
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}\;t \le -2.4305944631374538 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\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) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \sqrt[3]{\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) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\right) \cdot \sqrt[3]{\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) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \mathbf{elif}\;t \le -9.94161383116029 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \ell}{\frac{Om}{\ell}} \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;t \le -2.7013031257861712 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(\ell \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(\left(n \cdot \frac{1}{Om}\right) \cdot \ell\right) + \left((-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + t)_*\right))_*}\\ \mathbf{elif}\;t \le 5.689632516599635 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \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)}\\ \mathbf{elif}\;t \le 3.542707492116398 \cdot 10^{-81} \lor \neg \left(t \le 3.0534459407218795 \cdot 10^{+31}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{(\left(\ell \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left((-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + t)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(\ell \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(\left(n \cdot \frac{1}{Om}\right) \cdot \ell\right) + \left((-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + t)_*\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 5 regimes

  2. if t < -2.4305944631374538e+177

    1. Initial program 35.0

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

    3. Applied add-cube-cbrt35.2

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

    if -2.4305944631374538e+177 < t < -9.94161383116029e-42

    1. Initial program 30.2

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

    2. Taylor expanded around 0 35.3

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

    3. Simplified31.6

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

    5. Applied associate-*l*30.2

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

    if -9.94161383116029e-42 < t < -2.7013031257861712e-176 or 3.542707492116398e-81 < t < 3.0534459407218795e+31

    1. Initial program 30.9

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

    2. Taylor expanded around 0 36.4

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

    3. Simplified32.6

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

    5. Applied *-un-lft-identity32.6

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

    6. Applied associate-*r*32.6

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

    7. Simplified29.1

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

    9. Applied div-inv29.1

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

    10. Applied associate-*l*29.6

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

    if -2.7013031257861712e-176 < t < 5.689632516599635e-297

    1. Initial program 35.7

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

    3. Applied associate-*l*36.6

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

    if 5.689632516599635e-297 < t < 3.542707492116398e-81 or 3.0534459407218795e+31 < t

    1. Initial program 34.1

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

    2. Taylor expanded around 0 39.1

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

    3. Simplified35.3

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

    5. Applied *-un-lft-identity35.3

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

    6. Applied associate-*r*35.3

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

    7. Simplified31.8

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

    9. Applied sqrt-prod29.2

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

    10. Simplified29.2

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot n\right)}} \cdot \sqrt{(\left(\ell \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left((-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + t)_*\right))_*}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification30.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.4305944631374538 \cdot 10^{+177}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\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) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \sqrt[3]{\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) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\right) \cdot \sqrt[3]{\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) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \mathbf{elif}\;t \le -9.94161383116029 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \frac{n \cdot \ell}{\frac{Om}{\ell}} \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;t \le -2.7013031257861712 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(\ell \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(\left(n \cdot \frac{1}{Om}\right) \cdot \ell\right) + \left((-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + t)_*\right))_*}\\ \mathbf{elif}\;t \le 5.689632516599635 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \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)}\\ \mathbf{elif}\;t \le 3.542707492116398 \cdot 10^{-81} \lor \neg \left(t \le 3.0534459407218795 \cdot 10^{+31}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{(\left(\ell \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left((-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + t)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(\ell \cdot \left(\frac{U*}{Om} - \frac{U}{Om}\right)\right) \cdot \left(\left(n \cdot \frac{1}{Om}\right) \cdot \ell\right) + \left((-2 \cdot \left(\frac{\ell}{Om} \cdot \ell\right) + t)_*\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 5.7m)Debug logProfile

herbie shell --seed 2018252 +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*))))))