Average Error: 33.6 → 26.1
Time: 9.2m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)} \le 1.726385264769937 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)} \le 1.7781477494318306 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\\ \end{array}\]

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 (sqrt (* (* n 2) (* U (- (- t (/ (* 2 l) (/ Om l))) (* (- U U*) (* n (* (/ l Om) (/ l Om)))))))) < 1.726385264769937e-159 or 1.7781477494318306e+144 < (sqrt (* (* n 2) (* U (- (- t (/ (* 2 l) (/ Om l))) (* (- U U*) (* n (* (/ l Om) (/ l Om))))))))

    1. Initial program 51.8

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

    2. Initial simplification50.6

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

    4. Applied sqrt-prod49.5

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

    if 1.726385264769937e-159 < (sqrt (* (* n 2) (* U (- (- t (/ (* 2 l) (/ Om l))) (* (- U U*) (* n (* (/ l Om) (/ l Om)))))))) < 1.7781477494318306e+144

    1. Initial program 14.2

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

    2. Initial simplification12.6

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

    4. Applied associate-*l*5.2

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

    6. Applied associate-*l*1.1

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

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(U \cdot \left(\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)} \le 1.726385264769937 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;\sqrt{\left(U \cdot \left(\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)} \le 1.7781477494318306 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell \cdot 2}{\frac{Om}{\ell}}\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)}\\ \end{array}\]

Runtime

Time bar (total: 9.2m)Debug logProfile

herbie shell --seed 2018214 
(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*))))))