Average Error: 33.7 → 28.7
Time: 7.1m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3.688590472377885 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot 2}\\ \mathbf{elif}\;n \le 6.493394293397993 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot 2}\\ \end{array}\]

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 n < -3.688590472377885e+44 or 6.493394293397993e+24 < n

    1. Initial program 32.7

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

    2. Simplified32.7

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

    4. Applied associate-/l*30.7

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

    6. Applied associate-*l*29.9

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

    if -3.688590472377885e+44 < n < 6.493394293397993e+24

    1. Initial program 34.3

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

    2. Simplified34.3

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

    4. Applied associate-/l*31.3

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

    6. Applied associate-*l*28.1

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

  4. Final simplification28.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.688590472377885 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot 2}\\ \mathbf{elif}\;n \le 6.493394293397993 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(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*))))))