Average Error: 32.6 → 24.6
Time: 4.6m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
\[\begin{array}{l} \mathbf{if}\;U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \le -1.1075369580943968 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{if}\;U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \le -5.826898217003288 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left(\frac{\ell}{\frac{Om}{n}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)\right) \cdot \left(U - U*\right)\right) \cdot U\right)}\\ \mathbf{if}\;U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \le 4.537323238075 \cdot 10^{-313} \lor \neg \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \le 3.5255739183697707 \cdot 10^{+295}\right):\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

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 (* U (- t (* 2 (/ (pow l 2) Om)))) < -1.1075369580943968e+163 or -5.826898217003288e-228 < (* U (- t (* 2 (/ (pow l 2) Om)))) < 4.537323238075e-313 or 3.5255739183697707e+295 < (* U (- t (* 2 (/ (pow l 2) Om))))

    1. Initial program 43.1

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm
    3. Applied associate-/l*37.9

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

    if -1.1075369580943968e+163 < (* U (- t (* 2 (/ (pow l 2) Om)))) < -5.826898217003288e-228

    1. Initial program 22.6

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm
    3. Applied associate-/l*22.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm
    5. Applied associate-*l*16.4

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]
    6. Using strategy rm
    7. Applied add-cube-cbrt16.5

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\color{blue}{\left(\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right) \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\]
    8. Applied unpow-prod-down16.5

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left({\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot {\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\]
    9. Applied associate-*r*15.3

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2}\right)} \cdot \left(U - U*\right)\right)\right)}\]
    10. Applied simplify15.9

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

    if 4.537323238075e-313 < (* U (- t (* 2 (/ (pow l 2) Om)))) < 3.5255739183697707e+295

    1. Initial program 24.4

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm
    3. Applied associate-/l*24.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm
    5. Applied associate-*l*18.9

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]
    6. Using strategy rm
    7. Applied sqrt-prod11.5

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\]
  3. Recombined 3 regimes into one program.
  4. Applied simplify24.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \le -1.1075369580943968 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{if}\;U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \le -5.826898217003288 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left(\frac{\ell}{\frac{Om}{n}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)\right) \cdot \left(U - U*\right)\right) \cdot U\right)}\\ \mathbf{if}\;U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \le 4.537323238075 \cdot 10^{-313} \lor \neg \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \le 3.5255739183697707 \cdot 10^{+295}\right):\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}}\]

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed 2018206 
(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*))))))