Average Error: 34.6 → 32.4
Time: 30.8s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1.5437202692128072 \cdot 10^{-226} \lor \neg \left(n \le -7.178830274298204 \cdot 10^{-295}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \left({\left(\frac{1}{{\left({\ell}^{\left(-2\right)}\right)}^{1}}\right)}^{1} \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \left(n \cdot U\right)}{Om}\right) - 2 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right)}\\ \end{array}\]

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*

Derivation

  1. Split input into 2 regimes

  2. if n < -1.5437202692128072e-226 or -7.178830274298204e-295 < n

    1. Initial program 34.2

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

    2. Simplified31.9

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

    4. Applied associate-*r*31.6

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

    6. Applied associate-*r*31.8

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

    if -1.5437202692128072e-226 < n < -7.178830274298204e-295

    1. Initial program 39.5

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

    2. Simplified38.2

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

    4. Applied associate-*r*36.3

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

    6. Applied associate-*r*37.4

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

    7. Taylor expanded around inf 53.1

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(t \cdot \left(U \cdot n\right) + {\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{1}{\ell}\right)\right)}\right)}^{1}}\right)}^{1} \cdot \frac{U \cdot \left({n}^{2} \cdot U*\right)}{{Om}^{2}}\right) - 2 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)}}\]

    8. Simplified38.7

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right) + \left({\left(\frac{1}{{\left({\ell}^{\left(-2\right)}\right)}^{1}}\right)}^{1} \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \left(n \cdot U\right)}{Om}\right) - 2 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification32.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.5437202692128072 \cdot 10^{-226} \lor \neg \left(n \le -7.178830274298204 \cdot 10^{-295}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \left({\left(\frac{1}{{\left({\ell}^{\left(-2\right)}\right)}^{1}}\right)}^{1} \cdot \left(\frac{U*}{Om} \cdot \frac{n \cdot \left(n \cdot U\right)}{Om}\right) - 2 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))