Average Error: 33.0 → 25.2
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{2 \cdot \left((\left(\frac{n \cdot n}{\frac{Om}{U}}\right) \cdot \left(\frac{\ell \cdot \ell}{\frac{Om}{U*}}\right) + \left(U \cdot \left(t \cdot n\right)\right))_* - \left(\frac{U}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(n \cdot n\right) \cdot \left(\ell \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 5.4951726503445045 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot \frac{1}{\frac{Om}{\ell \cdot n}}\right) \cdot -2\right) \cdot (\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* + \left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \end{array}\]

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 (* (* (* 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

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

    3. Taylor expanded around inf 57.2

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

    4. Simplified44.9

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

    if 0.0 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 5.4951726503445045e+264

    1. Initial program 1.7

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

    if 5.4951726503445045e+264 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

    1. Initial program 57.5

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

    2. Initial simplification51.5

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

    4. Applied sub-neg51.5

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

    5. Applied distribute-rgt-in51.5

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

    6. Simplified44.8

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

    7. Taylor expanded around inf 44.0

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

    9. Applied associate-/l*42.8

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

    11. Applied div-inv42.8

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

  4. Final simplification25.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{2 \cdot \left((\left(\frac{n \cdot n}{\frac{Om}{U}}\right) \cdot \left(\frac{\ell \cdot \ell}{\frac{Om}{U*}}\right) + \left(U \cdot \left(t \cdot n\right)\right))_* - \left(\frac{U}{Om} \cdot \frac{U}{Om}\right) \cdot \left(\left(n \cdot n\right) \cdot \left(\ell \cdot \ell\right)\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 5.4951726503445045 \cdot 10^{+264}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot \frac{1}{\frac{Om}{\ell \cdot n}}\right) \cdot -2\right) \cdot (\left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* + \left(2 \cdot \left(U \cdot n\right)\right) \cdot t}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018242 +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*))))))