Average Error: 33.2 → 27.1
Time: 44.9s
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.4967444653905296 \cdot 10^{+178}:\\ \;\;\;\;{\left(\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot 2\right)\right) \cdot \left(-2 \cdot \ell - \left(n \cdot \left(U - U*\right)\right) \cdot \frac{\ell}{Om}\right) + t \cdot \left(n \cdot 2\right)\right) \cdot U\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\ell \le -2.7349420502749047 \cdot 10^{+50}:\\ \;\;\;\;\left|\sqrt{\left(\left(\left(U \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) + \left(U \cdot t\right) \cdot n\right) \cdot 2}\right|\\ \mathbf{elif}\;\ell \le -6.184123593935886 \cdot 10^{-118}:\\ \;\;\;\;{\left(\left(\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(U \cdot \left(n \cdot 2\right)\right) \cdot t\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\ell \le 9.160157018091963 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \le 1.0597865557101097 \cdot 10^{+47}:\\ \;\;\;\;\left|\sqrt{\left(\left(\left(U \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) + \left(U \cdot t\right) \cdot n\right) \cdot 2}\right|\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(U \cdot \left(n \cdot 2\right)\right) \cdot t\right)}^{\frac{1}{2}}\\ \end{array}\]

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 4 regimes

  2. if l < -1.4967444653905296e+178

    1. Initial program 60.9

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

    3. Applied *-un-lft-identity60.9

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

    4. Applied associate-*r*60.9

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

    5. Simplified49.6

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

    7. Applied sub-neg49.6

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

    8. Applied distribute-rgt-in49.6

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

    9. Simplified36.1

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

    11. Applied pow1/236.1

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

    13. Applied pow136.1

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

    14. Applied pow-pow36.1

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

    15. Simplified39.9

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

    if -1.4967444653905296e+178 < l < -2.7349420502749047e+50 or 9.160157018091963e-199 < l < 1.0597865557101097e+47

    1. Initial program 30.6

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

    3. Applied *-un-lft-identity30.6

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

    4. Applied associate-*r*30.6

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

    5. Simplified28.0

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

    7. Applied sub-neg28.0

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

    8. Applied distribute-rgt-in28.0

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

    9. Simplified24.2

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

    11. Applied add-sqr-sqrt24.2

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

    12. Applied rem-sqrt-square24.2

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

    13. Simplified27.2

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

    if -2.7349420502749047e+50 < l < -6.184123593935886e-118 or 1.0597865557101097e+47 < l

    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. Using strategy rm

    3. Applied *-un-lft-identity38.7

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

    4. Applied associate-*r*38.7

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

    5. Simplified33.9

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

    7. Applied sub-neg33.9

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

    8. Applied distribute-rgt-in33.9

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

    9. Simplified26.9

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

    11. Applied pow1/226.9

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

    if -6.184123593935886e-118 < l < 9.160157018091963e-199

    1. Initial program 23.6

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

    3. Applied associate-*l*24.2

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

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.4967444653905296 \cdot 10^{+178}:\\ \;\;\;\;{\left(\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot 2\right)\right) \cdot \left(-2 \cdot \ell - \left(n \cdot \left(U - U*\right)\right) \cdot \frac{\ell}{Om}\right) + t \cdot \left(n \cdot 2\right)\right) \cdot U\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\ell \le -2.7349420502749047 \cdot 10^{+50}:\\ \;\;\;\;\left|\sqrt{\left(\left(\left(U \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) + \left(U \cdot t\right) \cdot n\right) \cdot 2}\right|\\ \mathbf{elif}\;\ell \le -6.184123593935886 \cdot 10^{-118}:\\ \;\;\;\;{\left(\left(\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(U \cdot \left(n \cdot 2\right)\right) \cdot t\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\ell \le 9.160157018091963 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(U - U*\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right)\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \le 1.0597865557101097 \cdot 10^{+47}:\\ \;\;\;\;\left|\sqrt{\left(\left(\left(U \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) + \left(U \cdot t\right) \cdot n\right) \cdot 2}\right|\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(U \cdot 2\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-2 \cdot \ell - \left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) + \left(U \cdot \left(n \cdot 2\right)\right) \cdot t\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019052 
(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*))))))