Average Error: 32.9 → 26.7
Time: 5.8m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot n\right) \cdot U \le -2.4556303168677174 \cdot 10^{-237}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{if}\;\left(2 \cdot n\right) \cdot U \le 4.2298195104584 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \sqrt[3]{{\left(\left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}^{3}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\\ \end{array}\]

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* (* 2 n) U) < -2.4556303168677174e-237

    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 associate-*l*31.1

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

    5. Applied add-cube-cbrt31.7

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

    6. Taylor expanded around 0 31.7

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

    7. Applied simplify26.4

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

    9. Applied add-cube-cbrt27.1

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

    if -2.4556303168677174e-237 < (* (* 2 n) U) < 4.2298195104584e-318

    1. Initial program 52.7

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

    3. Applied associate-*l*40.2

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

    5. Applied add-cbrt-cube42.3

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

    6. Applied simplify41.4

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

    if 4.2298195104584e-318 < (* (* 2 n) U)

    1. Initial program 27.6

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

    3. Applied associate-*l*30.9

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

    5. Applied add-cube-cbrt31.4

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

    6. Taylor expanded around 0 31.4

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

    7. Applied simplify26.4

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

    9. Applied sqrt-prod18.7

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

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed 2018296 
(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*))))))