Average Error: 33.0 → 28.9
Time: 1.8m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.1866363658847081 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\sqrt{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(-\left(U - U*\right)\right) + \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_*\right))_* \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \cdot \sqrt{\sqrt{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(-\left(U - U*\right)\right) + \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_*\right))_* \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(-\left(U - U*\right)\right) + \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_*\right))_*} \cdot \sqrt{2 \cdot \left(U \cdot n\right)}\\ \end{array}\]

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 t < 1.1866363658847081e-298

    1. Initial program 33.2

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

    3. Applied unpow233.2

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

    4. Applied associate-*r*32.4

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

    6. Applied *-un-lft-identity32.4

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

    7. Applied associate-*r*32.4

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

    8. Simplified29.8

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

    10. Applied add-sqr-sqrt29.9

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

    if 1.1866363658847081e-298 < t

    1. Initial program 32.8

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

    3. Applied unpow232.8

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

    4. Applied associate-*r*32.0

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

    6. Applied *-un-lft-identity32.0

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

    7. Applied associate-*r*32.0

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

    8. Simplified29.5

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

    10. Applied sqrt-prod27.9

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

    11. Simplified27.9

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

  4. Final simplification28.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.1866363658847081 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\sqrt{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(-\left(U - U*\right)\right) + \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_*\right))_* \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \cdot \sqrt{\sqrt{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(-\left(U - U*\right)\right) + \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_*\right))_* \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right) \cdot \left(-\left(U - U*\right)\right) + \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_*\right))_*} \cdot \sqrt{2 \cdot \left(U \cdot n\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018256 +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*))))))