Average Error: 33.4 → 21.8
Time: 3.0m
Precision: 64
Internal Precision: 384
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \cdot \left(U + U\right) \le -7.61589945806054 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(n \cdot \left(\left(U + U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(\left(-\ell\right) + \left(-\ell\right)\right) - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{if}\;n \cdot \left(U + U\right) \le 1.7902863929574 \cdot 10^{-318}:\\ \;\;\;\;\sqrt{\left(\ell \cdot \left(\left(-2\right) - \left(U - U*\right) \cdot \frac{n}{Om}\right)\right) \cdot \left(\left(\ell \cdot \frac{n}{Om}\right) \cdot \left(U + U\right)\right) + \left(U \cdot \left(t \cdot n\right)\right) \cdot 2}\\ \mathbf{if}\;n \cdot \left(U + U\right) \le 2.883312708001724 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{if}\;n \cdot \left(U + U\right) \le 7.828326535659496 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(n \cdot \left(\left(U + U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(\left(-\ell\right) + \left(-\ell\right)\right) - \frac{n \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

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*

Derivation

  1. Split input into 3 regimes

  2. if (* n (+ U U)) < -7.61589945806054e-285 or 2.883312708001724e-214 < (* n (+ U U)) < 7.828326535659496e-64

    1. Initial program 27.2

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

    3. Applied associate-/l*23.8

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

    5. Applied sub-neg23.8

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

    6. Applied associate--l+23.8

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

    7. Applied distribute-lft-in23.8

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

    8. Applied simplify21.6

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

    10. Applied associate-*l*22.4

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

    if -7.61589945806054e-285 < (* n (+ U U)) < 1.7902863929574e-318

    1. Initial program 55.6

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

    3. Applied associate-/l*55.1

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

    5. Applied sub-neg55.1

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

    6. Applied associate--l+55.1

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

    7. Applied distribute-lft-in55.1

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

    8. Applied simplify54.0

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

    9. Taylor expanded around 0 47.6

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

    10. Applied simplify29.7

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

    if 1.7902863929574e-318 < (* n (+ U U)) < 2.883312708001724e-214 or 7.828326535659496e-64 < (* n (+ U U))

    1. Initial program 29.7

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

    3. Applied associate-/l*26.8

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

    5. Applied sqrt-prod15.7

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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*))))))