Average Error: 33.5 → 24.6
Time: 42.3s
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -5.3037709247372026 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\right) \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + t \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 3.384865516660387 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(U \cdot 2\right) \cdot \left(t \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\right) \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + t \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

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 n < -5.3037709247372026e+17 or 3.384865516660387e-160 < n

    1. Initial program 31.8

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

    3. Applied *-un-lft-identity31.8

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

    4. Applied associate-*r*31.8

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

    5. Simplified27.4

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

    7. Applied sub-neg27.4

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

    8. Applied distribute-rgt-in27.4

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

    9. Simplified25.1

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

    11. Applied add-cube-cbrt25.3

      \[\leadsto \sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \color{blue}{\left(\left(\sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)} \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right) \cdot \sqrt[3]{\left(2 \cdot U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)}\right)} \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\]

    if -5.3037709247372026e+17 < n < 3.384865516660387e-160

    1. Initial program 35.4

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

    3. Applied *-un-lft-identity35.4

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

    4. Applied associate-*r*35.4

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

    5. Simplified32.4

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

    7. Applied sub-neg32.4

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

    8. Applied distribute-rgt-in32.4

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

    9. Simplified25.8

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

    10. Taylor expanded around -inf 25.8

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

    11. Simplified23.9

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

  4. Final simplification24.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.3037709247372026 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\right) \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + t \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 3.384865516660387 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(U \cdot 2\right) \cdot \left(t \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\right) \cdot \sqrt[3]{\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + t \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(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*))))))