Average Error: 33.8 → 25.7
Time: 6.0m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.3204189403090777 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \le 5.382434498096398 \cdot 10^{-114}:\\ \;\;\;\;\left|\sqrt[3]{\left(2 \cdot n\right) \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{U \cdot \left(\sqrt{2} \cdot n\right)}} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{\sqrt[3]{\sqrt{2}} \cdot \left(\sqrt[3]{\sqrt[3]{U \cdot \left(\sqrt{2} \cdot n\right)}} \cdot \sqrt[3]{\sqrt[3]{U \cdot \left(\sqrt{2} \cdot n\right)}}\right)}\right)\\ \mathbf{elif}\;\ell \le 1.0408114946233754 \cdot 10^{+46}:\\ \;\;\;\;\left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\left(U \cdot \sqrt[3]{2 \cdot n}\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

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 3 regimes

  2. if l < -1.3204189403090777e+154 or 1.0408114946233754e+46 < l

    1. Initial program 52.6

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

    3. Applied *-un-lft-identity52.6

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

    4. Applied associate-*r*52.6

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

    5. Applied associate-/l*41.7

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

    6. Simplified41.7

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

    if -1.3204189403090777e+154 < l < 5.382434498096398e-114

    1. Initial program 26.8

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

    3. Applied add-cube-cbrt27.2

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

    4. Applied associate-*l*27.2

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

    5. Applied sqrt-prod21.0

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

    6. Simplified21.0

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

    8. Applied add-sqr-sqrt20.9

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

    9. Applied associate-*l*20.9

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

    10. Applied associate-*l*20.9

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

    11. Applied cbrt-prod20.9

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

    13. Applied add-cube-cbrt21.1

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

    14. Applied associate-*r*21.1

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

    15. Applied associate-*l*21.1

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

    16. Applied sqrt-prod19.3

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

    if 5.382434498096398e-114 < l < 1.0408114946233754e+46

    1. Initial program 28.9

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

    3. Applied add-cube-cbrt29.3

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

    4. Applied associate-*l*29.3

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

    5. Applied associate-*l*28.3

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

    6. Applied sqrt-prod23.1

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

    7. Simplified23.1

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.3204189403090777 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;\ell \le 5.382434498096398 \cdot 10^{-114}:\\ \;\;\;\;\left|\sqrt[3]{\left(2 \cdot n\right) \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{U \cdot \left(\sqrt{2} \cdot n\right)}} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{\sqrt[3]{\sqrt{2}} \cdot \left(\sqrt[3]{\sqrt[3]{U \cdot \left(\sqrt{2} \cdot n\right)}} \cdot \sqrt[3]{\sqrt[3]{U \cdot \left(\sqrt{2} \cdot n\right)}}\right)}\right)\\ \mathbf{elif}\;\ell \le 1.0408114946233754 \cdot 10^{+46}:\\ \;\;\;\;\left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\left(U \cdot \sqrt[3]{2 \cdot n}\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(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*))))))