Average Error: 33.1 → 24.1
Time: 3.1m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(n \cdot U\right) \cdot 2 \le -4.04674894421819 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot 0\right) + \left(\left(n \cdot U\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*}\\ \mathbf{if}\;\left(n \cdot U\right) \cdot 2 \le 1.3383398379117262 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*\right) \cdot U\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(-\left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right))_*}\\ \mathbf{if}\;\left(n \cdot U\right) \cdot 2 \le 6.6591248961963185 \cdot 10^{-146} \lor \neg \left(\left(n \cdot U\right) \cdot 2 \le 3.9166031812794964 \cdot 10^{+71}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot 0\right) + \left(\left(n \cdot U\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*}\\ \end{array}\]

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 (* 2 (* U n)) < -4.04674894421819e-309 or 6.6591248961963185e-146 < (* 2 (* U n)) < 3.9166031812794964e+71

    1. Initial program 25.5

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

    3. Applied add-sqr-sqrt48.8

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

    4. Applied prod-diff48.8

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

    5. Applied distribute-rgt-in48.8

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

    6. Applied simplify21.6

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

    7. Applied simplify20.8

      \[\leadsto \sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \color{blue}{\left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}}\]

    if -4.04674894421819e-309 < (* 2 (* U n)) < 1.3383398379117262e-303

    1. Initial program 55.9

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

    3. Applied add-sqr-sqrt58.8

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

    4. Applied prod-diff58.8

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

    5. Applied distribute-lft-in58.8

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

    6. Applied simplify55.5

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

    8. Applied associate-*l*37.7

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

    9. Applied simplify37.7

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

    if 1.3383398379117262e-303 < (* 2 (* U n)) < 6.6591248961963185e-146 or 3.9166031812794964e+71 < (* 2 (* U n))

    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 sqrt-prod21.0

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

  4. Applied simplify24.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\left(n \cdot U\right) \cdot 2 \le -4.04674894421819 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot 0\right) + \left(\left(n \cdot U\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*}\\ \mathbf{if}\;\left(n \cdot U\right) \cdot 2 \le 1.3383398379117262 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*\right) \cdot U\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(-\left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right))_*}\\ \mathbf{if}\;\left(n \cdot U\right) \cdot 2 \le 6.6591248961963185 \cdot 10^{-146} \lor \neg \left(\left(n \cdot U\right) \cdot 2 \le 3.9166031812794964 \cdot 10^{+71}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot 0\right) + \left(\left(n \cdot U\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right) + \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*}\\ \end{array}}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +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*))))))