Average Error: 33.3 → 27.0
Time: 2.5m
Precision: 64
Internal Precision: 384
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le -1.969335207900552 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(n \cdot \left(\frac{U}{Om} \cdot \ell\right)\right) \cdot \left(\ell \cdot 4\right)}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le -3.9936314212971254 \cdot 10^{-216}:\\ \;\;\;\;{\left(n \cdot \left(\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le 4.947851092670814 \cdot 10^{-87}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right) \cdot \sqrt[3]{U - U*}\right)}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le 1.879694727550129 \cdot 10^{+187}:\\ \;\;\;\;{\left(n \cdot \left(\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le 9.401278382498222 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{n \cdot \left(U + U\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(n \cdot \left(\frac{U}{Om} \cdot \ell\right)\right) \cdot \left(\ell \cdot 4\right)}\\ \end{array}\]

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 4 regimes

  2. if (* (+ U U) (- t (* (/ l Om) (+ l l)))) < -1.969335207900552e+297 or 9.401278382498222e+208 < (* (+ U U) (- t (* (/ l Om) (+ l l))))

    1. Initial program 47.8

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

    2. Taylor expanded around 0 47.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{0}\right)}\]

    3. Applied simplify44.4

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

    5. Applied pow1/244.4

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

    6. Taylor expanded around inf 54.9

      \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) - 4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}}^{\frac{1}{2}}\]

    7. Applied simplify39.6

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

    9. Applied div-inv39.6

      \[\leadsto \sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(\color{blue}{\left(n \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot 4\right)}\]

    10. Applied associate-*l*38.0

      \[\leadsto \sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \color{blue}{\left(n \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot U\right)\right)\right)} \cdot \left(\ell \cdot 4\right)}\]

    11. Applied simplify36.9

      \[\leadsto \sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(n \cdot \color{blue}{\left(\frac{U}{Om} \cdot \ell\right)}\right) \cdot \left(\ell \cdot 4\right)}\]

    if -1.969335207900552e+297 < (* (+ U U) (- t (* (/ l Om) (+ l l)))) < -3.9936314212971254e-216 or 4.947851092670814e-87 < (* (+ U U) (- t (* (/ l Om) (+ l l)))) < 1.879694727550129e+187

    1. Initial program 25.3

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

    2. Taylor expanded around 0 28.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{0}\right)}\]

    3. Applied simplify25.8

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

    5. Applied pow1/225.8

      \[\leadsto \color{blue}{{\left(\left(n \cdot \left(U + U\right)\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right)\right)}^{\frac{1}{2}}}\]
    6. Using strategy rm

    7. Applied associate-*l*18.7

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

    if -3.9936314212971254e-216 < (* (+ U U) (- t (* (/ l Om) (+ l l)))) < 4.947851092670814e-87

    1. Initial program 32.1

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

    3. Applied add-cube-cbrt32.1

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

    4. Applied associate-*r*32.1

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

    if 1.879694727550129e+187 < (* (+ U U) (- t (* (/ l Om) (+ l l)))) < 9.401278382498222e+208

    1. Initial program 27.3

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

    2. Taylor expanded around 0 28.0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{0}\right)}\]

    3. Applied simplify25.0

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

    5. Applied sqrt-prod38.9

      \[\leadsto \color{blue}{\sqrt{n \cdot \left(U + U\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)}}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(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*))))))