\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
Test:
Toniolo and Linder, Equation (13)
Bits:
128 bits
Bits error versus n
Bits error versus U
Bits error versus t
Bits error versus l
Bits error versus Om
Bits error versus U*
Time: 1.3 m
Input Error: 29.7
Output Error: 23.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\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)\right)\right)} & \text{when } \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right) \le 9.565553155495165 \cdot 10^{-196} \\ \sqrt[3]{{\left(\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\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)\right)}\right)}^3} & \text{when } \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right) \le 1.6564995191609956 \cdot 10^{+173} \\ \sqrt{\left(\left(t - \frac{{\ell}^2}{\frac{Om}{2}}\right) - \left(\frac{U}{Om} - \frac{U*}{Om}\right) \cdot \left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} & \text{when } \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right) \le 4.832796084473531 \cdot 10^{+232} \\ \sqrt{\left(\left(t - \frac{{\ell}^2}{\frac{Om}{2}}\right) - \left(\frac{U}{Om} - \frac{U*}{Om}\right) \cdot \left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)} & \text{when } \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right) \le 6.765140078268251 \cdot 10^{+298} \\ \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)} & \text{otherwise} \end{cases}\)

    if (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*)))) < 9.565553155495165e-196

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      43.0
    2. Using strategy rm
      43.0
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      43.0
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      41.9
    5. Using strategy rm
      41.9
    6. Applied associate-*l* to get
      \[\sqrt{\color{red}{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)\right)}}\]
      32.6

    if 9.565553155495165e-196 < (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*)))) < 1.6564995191609956e+173

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      1.5
    2. Using strategy rm
      1.5
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      1.5
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      1.5
    5. Using strategy rm
      1.5
    6. Applied add-cbrt-cube to get
      \[\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\right)}^3}}\]
      1.9

    if 1.6564995191609956e+173 < (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*)))) < 4.832796084473531e+232

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      0.9
    2. Using strategy rm
      0.9
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      0.9
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      0.9
    5. Applied taylor to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)\right)}\]
      15.5
    6. Taylor expanded around 0 to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{red}{\left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)}\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)}\right)}\]
      15.5
    7. Applied simplify to get
      \[\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)\right)}} \leadsto \color{blue}{\sqrt{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - \frac{\ell \cdot n}{\frac{Om}{\ell}} \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)}}\]
      7.7
    8. Applied simplify to get
      \[\sqrt{\color{red}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - \frac{\ell \cdot n}{\frac{Om}{\ell}} \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)}} \leadsto \sqrt{\color{blue}{\left(\left(t - \frac{{\ell}^2}{\frac{Om}{2}}\right) - \left(\frac{U}{Om} - \frac{U*}{Om}\right) \cdot \left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)}}\]
      7.7

    if 4.832796084473531e+232 < (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*)))) < 6.765140078268251e+298

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      0.7
    2. Using strategy rm
      0.7
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      0.7
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      0.7
    5. Applied taylor to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)\right)}\]
      15.3
    6. Taylor expanded around 0 to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{red}{\left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)}\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)}\right)}\]
      15.3
    7. Applied simplify to get
      \[\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{n \cdot \left({\ell}^2 \cdot U\right)}{{Om}^2} - \frac{n \cdot \left(U* \cdot {\ell}^2\right)}{{Om}^2}\right)\right)}} \leadsto \color{blue}{\sqrt{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - \frac{\ell \cdot n}{\frac{Om}{\ell}} \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)}}\]
      7.1
    8. Applied simplify to get
      \[\sqrt{\color{red}{\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - \frac{\ell \cdot n}{\frac{Om}{\ell}} \cdot \left(\frac{U}{Om} - \frac{U*}{Om}\right)\right)}} \leadsto \sqrt{\color{blue}{\left(\left(t - \frac{{\ell}^2}{\frac{Om}{2}}\right) - \left(\frac{U}{Om} - \frac{U*}{Om}\right) \cdot \left(\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)}}\]
      7.1

    if 6.765140078268251e+298 < (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*))))

    1. Started with
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{{\ell}^2}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      60.4
    2. Using strategy rm
      60.4
    3. Applied square-mult to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{red}{{\ell}^2}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      60.4
    4. Applied associate-/l* to get
      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{red}{\frac{\ell \cdot \ell}{Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)} \leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}\]
      51.6
    5. Using strategy rm
      51.6
    6. Applied sqrt-prod to get
      \[\color{red}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)\right)}} \leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^2\right) \cdot \left(U - U*\right)}}\]
      47.1
  1. Removed slow pow expressions

Original test:


(lambda ((n default) (U default) (t default) (l default) (Om default) (U* default))
  #:name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (sqr l) Om))) (* (* n (sqr (/ l Om))) (- U U*))))))