Average Error: 33.4 → 26.2
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -4.2293876163282466 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \left(n \cdot U\right)\right)\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \ell\right))_* + t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|{\left((\left((n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\left(2 \cdot U\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right)\right) + \left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right))_*\right)}^{\frac{1}{2}}\right|\\ \end{array}\]

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*

Derivation

  1. Split input into 2 regimes

  2. if U < -4.2293876163282466e-33

    1. Initial program 28.1

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

    2. Initial simplification25.6

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

    4. Applied sub-neg25.6

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

    5. Applied distribute-rgt-in25.6

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

    6. Simplified22.9

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

    if -4.2293876163282466e-33 < U

    1. Initial program 34.8

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

    2. Initial simplification34.1

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

    4. Applied sub-neg34.1

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

    5. Applied distribute-lft-in34.1

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

    6. Simplified29.5

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

    8. Applied add-sqr-sqrt29.5

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

    9. Applied rem-sqrt-square29.5

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

    10. Simplified27.0

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

    12. Applied pow1/227.0

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

  4. Final simplification26.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -4.2293876163282466 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{\left(\frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \left(n \cdot U\right)\right)\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(2 \cdot \ell\right))_* + t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|{\left((\left((n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\left(2 \cdot U\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right)\right) + \left(\left(2 \cdot n\right) \cdot \left(t \cdot U\right)\right))_*\right)}^{\frac{1}{2}}\right|\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018221 +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*))))))