Average Error: 33.3 → 29.0
Time: 5.1m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.2825366785368352 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \le -4.691191686917764 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n + \left(\frac{\left(U* \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{n} \cdot \frac{Om}{n}} - \left(\frac{U}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\left(-\ell \cdot 2\right) - \frac{\left(U - U*\right) \cdot n}{\frac{Om}{\ell}}\right) + t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \end{array}\]

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 n < -2.2825366785368352e-172

    1. Initial program 31.0

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

    2. Initial simplification30.2

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

    4. Applied associate-*r*29.7

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

    6. Applied associate-*l*26.9

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

    if -2.2825366785368352e-172 < n < -4.691191686917764e-193

    1. Initial program 36.5

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

    2. Initial simplification32.6

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

    3. Taylor expanded around -inf 50.7

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

    4. Simplified41.8

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

    if -4.691191686917764e-193 < n

    1. Initial program 34.4

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

    2. Initial simplification33.1

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

    4. Applied associate-*r*32.3

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

    6. Applied sub-neg32.3

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

    7. Applied associate--l+32.3

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

    8. Applied distribute-lft-in32.3

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

    9. Simplified29.7

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

  4. Final simplification29.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.2825366785368352 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(U \cdot n\right)\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;n \le -4.691191686917764 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot U\right) \cdot n + \left(\frac{\left(U* \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{n} \cdot \frac{Om}{n}} - \left(\frac{U}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot \left(U \cdot n\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\left(-\ell \cdot 2\right) - \frac{\left(U - U*\right) \cdot n}{\frac{Om}{\ell}}\right) + t \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 5.1m)Debug logProfile

herbie shell --seed 2018234 
(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*))))))