Average Error: 33.0 → 28.5
Time: 47.8s
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.025207585857326 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + 2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;t \le -8.292532043064404 \cdot 10^{-274}:\\ \;\;\;\;\sqrt[3]{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right))_*\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right))_*\right)}} \cdot \sqrt[3]{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right))_*\right)}}\right)\\ \mathbf{elif}\;t \le 1.3579639696583106 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\frac{n}{\frac{Om}{\ell}} \cdot \frac{U}{\frac{Om}{\ell}} - \frac{U*}{\frac{Om}{\ell}} \cdot \frac{n}{\frac{Om}{\ell}}\right))_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right))_*} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}\\ \end{array}\]

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 4 regimes

  2. if t < -8.025207585857326e-120

    1. Initial program 31.0

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

    3. Applied *-un-lft-identity31.0

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

    4. Applied associate-*r*31.0

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

    5. Simplified28.5

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

    7. Applied pow128.5

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

    8. Applied pow128.5

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

    9. Applied pow-prod-down28.5

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

    10. Simplified27.8

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

    if -8.025207585857326e-120 < t < -8.292532043064404e-274

    1. Initial program 35.0

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

    3. Applied *-un-lft-identity35.0

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

    4. Applied associate-*r*35.0

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

    5. Simplified32.6

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

    7. Applied add-cube-cbrt33.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right))_*\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right))_*\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right))_*\right)}}}\]

    if -8.292532043064404e-274 < t < 1.3579639696583106e+137

    1. Initial program 33.0

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

    3. Applied *-un-lft-identity33.0

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

    4. Applied associate-*r*33.0

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

    5. Simplified30.4

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

    6. Taylor expanded around 0 38.7

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

    7. Simplified29.3

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

    if 1.3579639696583106e+137 < t

    1. Initial program 36.2

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

    3. Applied *-un-lft-identity36.2

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

    4. Applied associate-*r*36.2

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

    5. Simplified33.9

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

    7. Applied sqrt-prod23.9

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

    8. Simplified23.9

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(U \cdot n\right)}} \cdot \sqrt{t - (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot 2\right) + \left(\left(U - U*\right) \cdot \left(n \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\right))_*}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification28.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.025207585857326 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t - \left(n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + 2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;t \le -8.292532043064404 \cdot 10^{-274}:\\ \;\;\;\;\sqrt[3]{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right))_*\right)}} \cdot \left(\sqrt[3]{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right))_*\right)}} \cdot \sqrt[3]{\sqrt{\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right))_*\right)}}\right)\\ \mathbf{elif}\;t \le 1.3579639696583106 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\left(t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\frac{n}{\frac{Om}{\ell}} \cdot \frac{U}{\frac{Om}{\ell}} - \frac{U*}{\frac{Om}{\ell}} \cdot \frac{n}{\frac{Om}{\ell}}\right))_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om}\right) \cdot \left(2 \cdot \ell\right) + \left(\left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right))_*} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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*))))))