Average Error: 33.3 → 25.7
Time: 2.5m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1.16578968560785 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

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 2 regimes

  2. if n < -1.16578968560785e-310

    1. Initial program 32.9

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

    3. Applied associate-/l*30.0

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

    5. Applied unpow230.0

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

    6. Applied associate-*r*29.3

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

    8. Applied associate-*l*29.6

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

    10. Applied add-sqr-sqrt29.8

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

    if -1.16578968560785e-310 < n

    1. Initial program 33.6

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

    3. Applied associate-/l*30.8

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

    5. Applied unpow230.8

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

    6. Applied associate-*r*30.0

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

    8. Applied associate-*l*29.3

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

    10. Applied sqrt-prod21.6

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.16578968560785 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018348 
(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*))))))