Average Error: 33.5 → 28.0
Time: 1.0m
Precision: 64
Internal Precision: 320
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.9310533439626915 \cdot 10^{-241}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\ell \cdot \left(n \cdot \frac{1}{Om}\right)\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(U \cdot -2\right)}{Om}\right) + \left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right))_*}\right|\\ \mathbf{elif}\;\ell \le -1.211398563110576 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\sqrt[3]{\frac{\ell}{Om} \cdot n} \cdot \left(\sqrt[3]{\frac{\ell}{Om} \cdot n} \cdot \sqrt[3]{\frac{\ell}{Om} \cdot n}\right)\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(U \cdot -2\right)}{Om}\right) + \left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right))_*}\right|\\ \end{array}\]

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 3 regimes

  2. if l < -1.9310533439626915e-241

    1. Initial program 35.3

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

    2. Initial simplification33.7

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

    4. Applied sub-neg33.7

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

    5. Applied distribute-rgt-in33.7

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

    6. Simplified30.0

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

    8. Applied add-sqr-sqrt30.0

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

    9. Applied rem-sqrt-square30.0

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

    10. Simplified29.2

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

    12. Applied div-inv29.2

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

    13. Applied associate-*l*28.5

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

    if -1.9310533439626915e-241 < l < -1.211398563110576e-253

    1. Initial program 29.2

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

    2. Initial simplification31.9

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

    4. Applied sqrt-prod39.3

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

    if -1.211398563110576e-253 < l

    1. Initial program 32.3

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

    2. Initial simplification31.2

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

    4. Applied sub-neg31.2

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

    5. Applied distribute-rgt-in31.2

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

    6. Simplified28.6

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

    8. Applied add-sqr-sqrt28.6

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

    9. Applied rem-sqrt-square28.6

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

    10. Simplified27.4

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

    12. Applied add-cube-cbrt27.4

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

  4. Final simplification28.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.9310533439626915 \cdot 10^{-241}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\ell \cdot \left(n \cdot \frac{1}{Om}\right)\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(U \cdot -2\right)}{Om}\right) + \left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right))_*}\right|\\ \mathbf{elif}\;\ell \le -1.211398563110576 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right))_*} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\sqrt[3]{\frac{\ell}{Om} \cdot n} \cdot \left(\sqrt[3]{\frac{\ell}{Om} \cdot n} \cdot \sqrt[3]{\frac{\ell}{Om} \cdot n}\right)\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(U \cdot -2\right)}{Om}\right) + \left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right))_*}\right|\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018257 +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*))))))