Average Error: 33.8 → 24.4
Time: 2.9m
Precision: 64
Internal Precision: 320
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(n \cdot U\right) \cdot 2} \cdot \sqrt{\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \le 8.737447431381124 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot 2} \cdot \sqrt{\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n \cdot U\right) \cdot 2\right) + \left(\left(-2\right) \cdot \ell - \left(\ell \cdot n\right) \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right) \cdot \left(2 \cdot \frac{U}{\frac{Om}{\ell \cdot n}}\right)}\\ \end{array}\]

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 (* (sqrt (* 2 (* U n))) (sqrt (- (- t (* (/ l Om) (* 2 l))) (* (* (- U U*) n) (* (/ l Om) (/ l Om)))))) < 8.737447431381124e+261

    1. Initial program 22.8

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

    2. Initial simplification19.3

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

    4. Applied sqrt-prod10.3

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

    if 8.737447431381124e+261 < (* (sqrt (* 2 (* U n))) (sqrt (- (- t (* (/ l Om) (* 2 l))) (* (* (- U U*) n) (* (/ l Om) (/ l Om))))))

    1. Initial program 39.5

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

    2. Initial simplification39.8

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

    3. Taylor expanded around 0 43.3

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

    4. Simplified38.0

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

    6. Applied sub-neg38.0

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

    7. Applied associate--l+38.0

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

    8. Applied distribute-lft-in38.0

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

    9. Simplified34.7

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

    10. Taylor expanded around 0 32.4

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

    12. Applied associate-/l*31.7

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

  4. Final simplification24.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(n \cdot U\right) \cdot 2} \cdot \sqrt{\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \le 8.737447431381124 \cdot 10^{+261}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot 2} \cdot \sqrt{\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(n \cdot U\right) \cdot 2\right) + \left(\left(-2\right) \cdot \ell - \left(\ell \cdot n\right) \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right) \cdot \left(2 \cdot \frac{U}{\frac{Om}{\ell \cdot n}}\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018217 
(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*))))))