Average Error: 33.1 → 22.9
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(n \cdot 2\right) \cdot U \le -7.208495143851399 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\left(n \cdot 2\right) \cdot U \le 4.78909564051676 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot n\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\\ \end{array}\]

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) < -7.208495143851399e-231

    1. Initial program 27.1

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

    3. Applied associate-/l*23.9

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

    5. Applied add-cube-cbrt24.0

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

    6. Applied unpow-prod-down24.0

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

    7. Applied associate-*r*23.1

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

    if -7.208495143851399e-231 < (* (* 2 n) U) < 4.78909564051676e-310

    1. Initial program 51.2

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

    3. Applied associate-/l*50.0

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

    5. Applied associate-*l*36.3

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

    7. Applied add-cube-cbrt36.4

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

    8. Applied unpow-prod-down36.4

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

    9. Applied associate-*r*34.8

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

    if 4.78909564051676e-310 < (* (* 2 n) U)

    1. Initial program 27.9

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

    3. Applied associate-/l*24.8

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

    5. Applied sqrt-prod15.9

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

  4. Final simplification22.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(n \cdot 2\right) \cdot U \le -7.208495143851399 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot 2\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot n\right)\right)\right)}\\ \mathbf{elif}\;\left(n \cdot 2\right) \cdot U \le 4.78909564051676 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left({\left(\sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot \left({\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)}^{2} \cdot n\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018215 
(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*))))))