Average Error: 33.4 → 26.7
Time: 4.0m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(U \cdot 2\right) \cdot n \le -5.662901191691462 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{(\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot \left(-2\right)\right) + \left((\left(n \cdot \left(U* - U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) + t)_*\right))_* \cdot \left(2 \cdot \left(U \cdot n\right)\right) + \left(U \cdot 0\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{if}\;\left(U \cdot 2\right) \cdot n \le 2.5178784826924444 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

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 (* (* U 2) n) < -5.662901191691462e-282

    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. Using strategy rm

    3. Applied associate-*l*31.6

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

    5. Applied associate-/l*29.2

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

    7. Applied add-sqr-sqrt59.4

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

    8. Applied prod-diff59.4

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

    9. Applied distribute-lft-in59.4

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

    10. Applied distribute-rgt-in59.4

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

    11. Applied simplify27.9

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

    12. Applied simplify27.0

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

    if -5.662901191691462e-282 < (* (* U 2) n) < 2.5178784826924444e-294

    1. Initial program 52.9

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

    3. Applied associate-*l*40.5

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

    5. Applied associate-/l*37.3

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

    if 2.5178784826924444e-294 < (* (* U 2) n)

    1. Initial program 27.9

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

    3. Applied sqrt-prod20.5

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

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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*))))))