Average Error: 33.3 → 27.6
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -5.83102707045725 \cdot 10^{-138} \lor \neg \left(\ell \le 1.8366604876334384 \cdot 10^{-128}\right):\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(n \cdot \left(\sqrt[3]{\frac{\ell}{Om}} \cdot \left(\left(U \cdot -2\right) \cdot \left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(\ell \cdot n\right) \cdot \left(U \cdot -2\right)}{Om}\right) + \left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right))_*}\right|\\ \end{array}\]

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 l < -5.83102707045725e-138 or 1.8366604876334384e-128 < l

    1. Initial program 38.1

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

    2. Initial simplification34.5

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

    4. Applied sub-neg34.5

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

    5. Applied distribute-rgt-in34.5

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

    6. Simplified30.7

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

    8. Applied associate-*l*28.5

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

    10. Applied add-cube-cbrt28.7

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

    11. Applied associate-*r*28.7

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

    if -5.83102707045725e-138 < l < 1.8366604876334384e-128

    1. Initial program 24.7

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

    2. Initial simplification26.7

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

    4. Applied sub-neg26.7

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

    5. Applied distribute-rgt-in26.7

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

    6. Simplified26.4

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

    8. Applied add-sqr-sqrt26.4

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

    9. Applied rem-sqrt-square26.4

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

    10. Simplified25.6

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

  4. Final simplification27.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -5.83102707045725 \cdot 10^{-138} \lor \neg \left(\ell \le 1.8366604876334384 \cdot 10^{-128}\right):\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(n \cdot \left(\sqrt[3]{\frac{\ell}{Om}} \cdot \left(\left(U \cdot -2\right) \cdot \left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(\ell \cdot n\right) \cdot \left(U \cdot -2\right)}{Om}\right) + \left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right))_*}\right|\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018348 +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*))))))