Average Error: 33.6 → 27.8
Time: 1.0m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.9794942304657505 \cdot 10^{-168}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(-2 \cdot U\right)}{Om}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \mathbf{elif}\;t \le 2.7248217525227904 \cdot 10^{-282}:\\ \;\;\;\;{\left(t \cdot \left(\left(U \cdot n\right) \cdot 2\right) + (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(\left(\left(-2 \cdot U\right) \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 4.601369894603888 \cdot 10^{-157} \lor \neg \left(t \le 2.379629704185496 \cdot 10^{+103}\right):\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(-2 \cdot U\right)}{Om}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \end{array}\]

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 3 regimes

  2. if t < -4.9794942304657505e-168 or 4.601369894603888e-157 < t < 2.379629704185496e+103

    1. Initial program 32.0

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

    2. Initial simplification31.0

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

    4. Applied sub-neg31.0

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

    5. Applied distribute-rgt-in31.0

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

    6. Simplified28.0

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

    8. Applied add-sqr-sqrt28.0

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

    9. Applied rem-sqrt-square28.0

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

    10. Simplified26.8

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

    if -4.9794942304657505e-168 < t < 2.7248217525227904e-282

    1. Initial program 36.0

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

    2. Initial simplification34.2

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

    4. Applied sub-neg34.2

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

    5. Applied distribute-rgt-in34.2

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

    6. Simplified30.2

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

    8. Applied associate-*l*29.0

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

    10. Applied pow1/229.0

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

    if 2.7248217525227904e-282 < t < 4.601369894603888e-157 or 2.379629704185496e+103 < t

    1. Initial program 36.1

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

    2. Initial simplification33.8

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

    4. Applied sqrt-prod29.4

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

  4. Final simplification27.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.9794942304657505 \cdot 10^{-168}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(-2 \cdot U\right)}{Om}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \mathbf{elif}\;t \le 2.7248217525227904 \cdot 10^{-282}:\\ \;\;\;\;{\left(t \cdot \left(\left(U \cdot n\right) \cdot 2\right) + (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* \cdot \left(\left(\left(-2 \cdot U\right) \cdot \frac{\ell}{Om}\right) \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 4.601369894603888 \cdot 10^{-157} \lor \neg \left(t \le 2.379629704185496 \cdot 10^{+103}\right):\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*} \cdot \sqrt{\left(U \cdot n\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(n \cdot \frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(n \cdot \ell\right) \cdot \left(-2 \cdot U\right)}{Om}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018250 +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*))))))