Average Error: 32.6 → 26.1
Time: 4.2m
Precision: 64
Internal precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;U \le -1.7106492987135438 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t + \left(\left(U \cdot n\right) \cdot \left(\ell + \ell\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) - \frac{U - U*}{\frac{Om}{n}}\right)\right)}\\ \mathbf{if}\;U \le 7.011714412203457 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t + \left(\ell \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) - \left(U - U*\right) \cdot \frac{n}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(n + n\right) \cdot U\right) \cdot t + \left(\left(U \cdot n\right) \cdot \left(\ell + \ell\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\left(-2\right) - \frac{U - U*}{\frac{Om}{n}}\right)\right)}\\ \end{array}\]

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*

Derivation

  1. Split input into 2 regimes.

  2. if U < -1.7106492987135438e+40 or 7.011714412203457e-69 < U

    1. Initial program 26.6

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

    2. Applied taylor 30.3

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

    3. Taylor expanded around 0 30.3

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

    4. Applied simplify 24.1

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

    6. Applied sub-neg 24.1

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

    7. Applied associate--l+ 24.1

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

    8. Applied distribute-lft-in 24.1

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

    9. Applied simplify 23.2

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

    if -1.7106492987135438e+40 < U < 7.011714412203457e-69

    1. Initial program 36.8

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

    2. Applied taylor 41.1

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

    3. Taylor expanded around 0 41.1

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

    4. Applied simplify 35.6

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

    6. Applied associate-*l* 31.3

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

    8. Applied sub-neg 31.3

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

    9. Applied associate--l+ 31.3

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

    10. Applied distribute-lft-in 31.3

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

    11. Applied simplify 28.1

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

Runtime

Time bar (total: 4.2m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(836365947 3039842136 2196419551 1375617809 2119087278 3151035938)'
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*))))))