Average Error: 33.0 → 24.7
Time: 4.0m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{\left(n \cdot \left(U \cdot t\right) + \left(\frac{\left(U* \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{n} \cdot \frac{Om}{n}} - \left(\frac{U}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right) + \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)\right) \cdot \left(\left(-2 \cdot \ell\right) - \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array}\]

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 0.0

    1. Initial program 56.9

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

    2. Initial simplification55.0

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

    3. Taylor expanded around -inf 56.1

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

    4. Simplified52.6

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

    5. Taylor expanded around -inf 57.2

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

    6. Simplified43.7

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

    if 0.0 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

    1. Initial program 28.2

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

    2. Initial simplification27.2

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

    3. Taylor expanded around -inf 34.5

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

    4. Simplified27.8

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

    6. Applied sub-neg27.8

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

    7. Applied associate--l+27.8

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

    8. Applied distribute-rgt-in27.8

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

    9. Simplified25.6

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

    11. Applied associate-*l*24.6

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

    12. Taylor expanded around -inf 25.0

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

    13. Simplified20.8

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

  4. Final simplification24.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{\left(n \cdot \left(U \cdot t\right) + \left(\frac{\left(U* \cdot U\right) \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{n} \cdot \frac{Om}{n}} - \left(\frac{U}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(n \cdot n\right)\right)\right)\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(U \cdot n\right) \cdot 2\right) + \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)\right) \cdot \left(\left(-2 \cdot \ell\right) - \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}\right)}\\ \end{array}\]

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2018242 
(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*))))))