Average Error: 33.4 → 26.7
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\ell}{Om} \le -8.720869404114973 \cdot 10^{-11} \lor \neg \left(\frac{\ell}{Om} \le 8.98361901046087 \cdot 10^{-91}\right):\\ \;\;\;\;\sqrt{\sqrt{\left(\left(\frac{\ell}{Om} \cdot \left(-2 \cdot U\right)\right) \cdot n\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_* + t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(\left(\frac{\ell}{Om} \cdot \left(-2 \cdot U\right)\right) \cdot n\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_* + t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\frac{\left(\ell \cdot n\right) \cdot \left(-2 \cdot U\right)}{Om}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \end{array}\]

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 2 regimes

  2. if (/ l Om) < -8.720869404114973e-11 or 8.98361901046087e-91 < (/ l Om)

    1. Initial program 44.7

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

    2. Initial simplification39.9

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

    4. Applied sub-neg39.9

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

    5. Applied distribute-rgt-in39.9

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

    6. Simplified32.9

      \[\leadsto \sqrt{t \cdot \left(2 \cdot \left(U \cdot n\right)\right) + \color{blue}{\left(\left(n \cdot \left(U \cdot -2\right)\right) \cdot \frac{\ell}{Om}\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_*}}\]
    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 add-sqr-sqrt29.2

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

    if -8.720869404114973e-11 < (/ l Om) < 8.98361901046087e-91

    1. Initial program 24.6

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

    2. Initial simplification25.6

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

    4. Applied sub-neg25.6

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

    5. Applied distribute-rgt-in25.6

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

    6. Simplified25.7

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

    8. Applied add-sqr-sqrt25.7

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

    9. Applied rem-sqrt-square25.7

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

    10. Simplified24.8

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

  4. Final simplification26.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\ell}{Om} \le -8.720869404114973 \cdot 10^{-11} \lor \neg \left(\frac{\ell}{Om} \le 8.98361901046087 \cdot 10^{-91}\right):\\ \;\;\;\;\sqrt{\sqrt{\left(\left(\frac{\ell}{Om} \cdot \left(-2 \cdot U\right)\right) \cdot n\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_* + t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(\left(\frac{\ell}{Om} \cdot \left(-2 \cdot U\right)\right) \cdot n\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_* + t \cdot \left(\left(U \cdot n\right) \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\frac{\left(\ell \cdot n\right) \cdot \left(-2 \cdot U\right)}{Om}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018274 +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*))))))