Average Error: 33.3 → 27.7
Time: 3.2m
Precision: 64
Internal Precision: 384
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right) = -\infty:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right) \cdot \left(4 \cdot \ell\right)}\\ \mathbf{if}\;t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right) \le 3.437860867674477 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{if}\;t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right) \le 9.025178386511885 \cdot 10^{+265}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right) \cdot \left(4 \cdot \ell\right)}\\ \end{array}\]

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*

Derivation

  1. Split input into 3 regimes

  2. if (- t (* (/ l Om) (+ l l))) or 9.025178386511885e+265 < (- t (* (/ l Om) (+ l l)))

    1. Initial program 57.0

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

    2. Taylor expanded around 0 56.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{0}\right)}\]

    3. Applied simplify54.6

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

    5. Applied pow1/254.6

      \[\leadsto \color{blue}{{\left(\left(\left(n + n\right) \cdot U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right)\right)}^{\frac{1}{2}}}\]

    6. Taylor expanded around inf 53.5

      \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) - 4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}}^{\frac{1}{2}}\]

    7. Applied simplify40.3

      \[\leadsto \color{blue}{\sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(\frac{n}{Om} \cdot \left(U \cdot \ell\right)\right) \cdot \left(4 \cdot \ell\right)}}\]

    if (- t (* (/ l Om) (+ l l))) < 3.437860867674477e-207

    1. Initial program 28.9

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

    3. Applied associate-*l*28.8

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

    5. Applied associate-/l*25.6

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

    if 3.437860867674477e-207 < (- t (* (/ l Om) (+ l l))) < 9.025178386511885e+265

    1. Initial program 27.2

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

    3. Applied sqrt-prod24.0

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

Runtime

Time bar (total: 3.2m)Debug log

herbie shell --seed '#(2479486159 2123901208 2662424940 349789437 14252662 202027171)' 
(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*))))))