Average Error: 33.8 → 22.4
Time: 1.4m
Precision: 64
Internal Precision: 320
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\left(n \cdot \left(-2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)\right) \cdot \left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\ell \cdot 2\right))_*\right) + \left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right))_* \le 5.7448471101483 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_*} \cdot \sqrt{\left(n \cdot U\right) \cdot 2}\\ \mathbf{elif}\;(\left(\left(n \cdot \left(-2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)\right) \cdot \left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\ell \cdot 2\right))_*\right) + \left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right))_* \le 9.099643860774243 \cdot 10^{+305}:\\ \;\;\;\;\left|\sqrt{(\left(\left(n \cdot \left(-2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)\right) \cdot \left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\ell \cdot 2\right))_*\right) + \left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right))_*}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_*} \cdot \sqrt{\left(n \cdot U\right) \cdot 2}\\ \end{array}\]

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 (fma (* (* n 2) (* (/ l Om) (- U))) (fma (- U U*) (* n (/ l Om)) (* 2 l)) (* (* n 2) (* t U))) < 5.7448471101483e-319 or 9.099643860774243e+305 < (fma (* (* n 2) (* (/ l Om) (- U))) (fma (- U U*) (* n (/ l Om)) (* 2 l)) (* (* n 2) (* t U)))

    1. Initial program 50.9

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

    2. Initial simplification51.2

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

    4. Applied sqrt-prod48.9

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

    if 5.7448471101483e-319 < (fma (* (* n 2) (* (/ l Om) (- U))) (fma (- U U*) (* n (/ l Om)) (* 2 l)) (* (* n 2) (* t U))) < 9.099643860774243e+305

    1. Initial program 20.1

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

    2. Initial simplification18.1

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

    4. Applied sub-neg18.1

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

    5. Applied distribute-rgt-in18.1

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

    6. Simplified12.4

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

    8. Applied associate-*l*9.3

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

    10. Applied add-cube-cbrt9.5

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

    12. Applied add-sqr-sqrt9.5

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

    13. Applied rem-sqrt-square9.5

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

    14. Simplified1.2

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;(\left(\left(n \cdot \left(-2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)\right) \cdot \left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\ell \cdot 2\right))_*\right) + \left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right))_* \le 5.7448471101483 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_*} \cdot \sqrt{\left(n \cdot U\right) \cdot 2}\\ \mathbf{elif}\;(\left(\left(n \cdot \left(-2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)\right) \cdot \left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\ell \cdot 2\right))_*\right) + \left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right))_* \le 9.099643860774243 \cdot 10^{+305}:\\ \;\;\;\;\left|\sqrt{(\left(\left(n \cdot \left(-2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)\right) \cdot \left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\ell \cdot 2\right))_*\right) + \left(\left(n \cdot 2\right) \cdot \left(t \cdot U\right)\right))_*}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(\frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right))_*} \cdot \sqrt{\left(n \cdot U\right) \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(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*))))))