Average Error: 33.4 → 28.1
Time: 5.0m
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}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le -5.9304744781634494 \cdot 10^{-266}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 2.2604837274481 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(U \cdot t\right) \cdot n + \frac{\left(n \cdot n\right) \cdot \left(U* \cdot U\right)}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}\right) + \frac{\left(-U\right) \cdot U}{\frac{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}{n \cdot n}}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 3.989463810694392 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < -5.9304744781634494e-266 or 3.989463810694392e+287 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

    1. Initial program 59.4

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

    2. Initial simplification52.9

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

    4. Applied associate-*r*51.3

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

    6. Applied add-cube-cbrt51.5

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

    if -5.9304744781634494e-266 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 2.2604837274481e-318

    1. Initial program 56.4

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

    2. Initial simplification56.3

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

    3. Taylor expanded around inf 57.0

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

    4. Simplified42.7

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

    if 2.2604837274481e-318 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 3.989463810694392e+287

    1. Initial program 1.6

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

  4. Final simplification28.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le -5.9304744781634494 \cdot 10^{-266}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 2.2604837274481 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(U \cdot t\right) \cdot n + \frac{\left(n \cdot n\right) \cdot \left(U* \cdot U\right)}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}\right) + \frac{\left(-U\right) \cdot U}{\frac{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}{n \cdot n}}\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 3.989463810694392 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array}\]

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2018219 
(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*))))))