Average Error: 33.1 → 26.5
Time: 45.7s
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le 1.0955796777959 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\sqrt[3]{U \cdot \left(\left(n \cdot \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2\right)}} \cdot \sqrt{\sqrt[3]{U \cdot \left(\left(n \cdot \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2\right)} \cdot \sqrt[3]{U \cdot \left(\left(n \cdot \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(n \cdot \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2}\\ \end{array}\]

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 2 regimes

  2. if U < 1.0955796777959e-310

    1. Initial program 32.4

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

    3. Applied pow132.4

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

    4. Applied pow132.4

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

    5. Applied pow-prod-down32.4

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

    6. Simplified32.0

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

    8. Applied associate-/l*29.1

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

    10. Applied add-cube-cbrt29.5

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

    11. Applied sqrt-prod29.5

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

    if 1.0955796777959e-310 < U

    1. Initial program 33.7

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

    3. Applied pow133.7

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

    4. Applied pow133.7

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

    5. Applied pow-prod-down33.7

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

    6. Simplified33.2

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

    8. Applied associate-/l*30.8

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

    10. Applied unpow-prod-down30.8

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

    11. Applied sqrt-prod23.6

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

    12. Simplified23.6

      \[\leadsto \color{blue}{\sqrt{U}} \cdot \sqrt{{\left(\left(\left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot n\right) \cdot 2\right)}^{1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification26.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 1.0955796777959 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\sqrt[3]{U \cdot \left(\left(n \cdot \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2\right)}} \cdot \sqrt{\sqrt[3]{U \cdot \left(\left(n \cdot \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2\right)} \cdot \sqrt[3]{U \cdot \left(\left(n \cdot \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(n \cdot \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_* - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +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*))))))