Average Error: 33.4 → 27.2
Time: 55.0s
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3.38239343875157 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot \left(n \cdot \left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

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 2 regimes

  2. if n < -3.38239343875157e-310

    1. Initial program 33.4

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

    3. Applied associate-*l*33.5

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

    5. Applied associate-/l*30.8

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

    7. Applied add-cube-cbrt30.8

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

    8. Applied associate-*r*30.8

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

    if -3.38239343875157e-310 < n

    1. Initial program 33.4

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

    3. Applied associate-*l*33.6

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

    5. Applied associate-/l*31.0

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

    7. Applied sqrt-prod23.6

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

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3.38239343875157 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot \left(n \cdot \left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{2}}\right)\right)\right)\right) \cdot U\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +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*))))))