Average Error: 32.8 → 30.6
Time: 5.1m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 8.757359481263745 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{2 \cdot \left(U \cdot n\right)}\\ \end{array}\]

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 t < 8.757359481263745e-262

    1. Initial program 33.2

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

    2. Initial simplification32.2

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

    4. Applied associate-*r*31.5

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

    if 8.757359481263745e-262 < t

    1. Initial program 32.4

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

    2. Initial simplification30.7

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

    4. Applied sqrt-prod29.5

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

  4. Final simplification30.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 8.757359481263745 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \frac{\ell}{Om} \cdot \left(\left(\left(U - U*\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(2 \cdot \left(U \cdot n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) - \left(\left(U - U*\right) \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{2 \cdot \left(U \cdot n\right)}\\ \end{array}\]

Runtime

Time bar (total: 5.1m)Debug logProfile

herbie shell --seed 2018220 
(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*))))))