Average Error: 33.2 → 28.3
Time: 53.0s
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U* \le -3.4671120923327836 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + \left(\left(\frac{\ell}{Om} \cdot \left(U \cdot -2\right)\right) \cdot n\right) \cdot (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(\ell \cdot U\right) + \left(2 \cdot \ell\right))_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + \left(\sqrt[3]{\left(\frac{\ell}{Om} \cdot \left(U \cdot -2\right)\right) \cdot n} \cdot \left(\sqrt[3]{\left(\frac{\ell}{Om} \cdot \left(U \cdot -2\right)\right) \cdot n} \cdot \sqrt[3]{\left(\frac{\ell}{Om} \cdot \left(U \cdot -2\right)\right) \cdot n}\right)\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}\\ \end{array}\]

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 U* < -3.4671120923327836e+142

    1. Initial program 34.8

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

    2. Initial simplification38.5

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

    4. Applied sub-neg38.5

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

    5. Applied distribute-rgt-in38.5

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

    6. Simplified36.6

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

    8. Applied associate-*l*35.1

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

    9. Taylor expanded around 0 37.7

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

    10. Simplified36.0

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

    if -3.4671120923327836e+142 < U*

    1. Initial program 33.0

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

    2. Initial simplification31.1

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

    4. Applied sub-neg31.1

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

    5. Applied distribute-rgt-in31.1

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

    6. Simplified28.1

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

    8. Applied associate-*l*26.9

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

    10. Applied add-cube-cbrt27.0

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

  4. Final simplification28.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;U* \le -3.4671120923327836 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + \left(\left(\frac{\ell}{Om} \cdot \left(U \cdot -2\right)\right) \cdot n\right) \cdot (\left(\ell \cdot U*\right) \cdot \left(\frac{-n}{Om}\right) + \left((\left(\frac{n}{Om}\right) \cdot \left(\ell \cdot U\right) + \left(2 \cdot \ell\right))_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot t + \left(\sqrt[3]{\left(\frac{\ell}{Om} \cdot \left(U \cdot -2\right)\right) \cdot n} \cdot \left(\sqrt[3]{\left(\frac{\ell}{Om} \cdot \left(U \cdot -2\right)\right) \cdot n} \cdot \sqrt[3]{\left(\frac{\ell}{Om} \cdot \left(U \cdot -2\right)\right) \cdot n}\right)\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 53.0s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes28.128.319.09.1-1.1%
herbie shell --seed 2018352 +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*))))))