Average Error: 33.4 → 28.5
Time: 3.0m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.403807017722621 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\left(2 \cdot \frac{U \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(\left(-2\right) \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) + \left(\left(U \cdot n\right) \cdot 2\right) \cdot t}\\ \mathbf{elif}\;\ell \le 6.118504296871457 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\frac{\ell}{Om} \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) \cdot \left(\left(-2\right) \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) + \left(\left(U \cdot n\right) \cdot 2\right) \cdot t}} \cdot \sqrt{\sqrt{\left(\frac{\ell}{Om} \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) \cdot \left(\left(-2\right) \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) + \left(\left(U \cdot n\right) \cdot 2\right) \cdot t}}\\ \end{array}\]

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 l < -1.403807017722621e-126

    1. Initial program 38.7

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

    2. Initial simplification34.9

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

    4. Applied associate-*r*34.0

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

    6. Applied sub-neg34.0

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

    7. Applied associate--l+34.0

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

    8. Applied distribute-lft-in34.0

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

    9. Simplified30.5

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

    10. Taylor expanded around 0 30.4

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

    if -1.403807017722621e-126 < l < 6.118504296871457e-297

    1. Initial program 23.8

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

    2. Initial simplification27.1

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

    4. Applied associate-*l*23.8

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

    if 6.118504296871457e-297 < l

    1. Initial program 33.8

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

    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. Using strategy rm

    4. Applied associate-*r*31.9

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

    6. Applied sub-neg31.9

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

    7. Applied associate--l+31.9

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

    8. Applied distribute-lft-in31.9

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

    9. Simplified29.1

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

    11. Applied add-sqr-sqrt29.3

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

  4. Final simplification28.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.403807017722621 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\left(2 \cdot \frac{U \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(\left(-2\right) \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) + \left(\left(U \cdot n\right) \cdot 2\right) \cdot t}\\ \mathbf{elif}\;\ell \le 6.118504296871457 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot \left(\left(t - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(\frac{\ell}{Om} \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) \cdot \left(\left(-2\right) \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) + \left(\left(U \cdot n\right) \cdot 2\right) \cdot t}} \cdot \sqrt{\sqrt{\left(\frac{\ell}{Om} \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) \cdot \left(\left(-2\right) \cdot \ell - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) + \left(\left(U \cdot n\right) \cdot 2\right) \cdot t}}\\ \end{array}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2018221 
(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*))))))