Average Error: 33.3 → 29.1
Time: 9.4m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le 6.6586973867411 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(\frac{\left(\ell \cdot \ell\right) \cdot -2}{Om} + t\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\left(\frac{\left(\ell \cdot \ell\right) \cdot -2}{Om} + t\right) - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array}\]

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 < 6.6586973867411e-311

    1. Initial program 33.3

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

    2. Initial simplification33.5

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

    4. Applied associate-*r*32.5

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

    6. Applied associate-*l*32.4

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

    if 6.6586973867411e-311 < n

    1. Initial program 33.3

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

    2. Initial simplification33.8

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

    4. Applied associate-*r*32.7

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

    6. Applied sqrt-prod25.8

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

  4. Final simplification29.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le 6.6586973867411 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(\left(\frac{\left(\ell \cdot \ell\right) \cdot -2}{Om} + t\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\left(\frac{\left(\ell \cdot \ell\right) \cdot -2}{Om} + t\right) - \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 9.4m)Debug logProfile

herbie shell --seed 2018234 
(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*))))))