Average Error: 33.8 → 27.1
Time: 4.7m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \le 4.2799990785414 \cdot 10^{-313}:\\ \;\;\;\;\sqrt{\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U* - U\right) + \left(t - \frac{2}{Om} \cdot \left(\ell \cdot \ell\right)\right))_* + \left(n \cdot 2\right) \cdot \left(U \cdot 0\right)}\\ \mathbf{if}\;\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \le 3.4777793799200777 \cdot 10^{+248}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot \sqrt{(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left((\left(n \cdot \left(U* - U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) + t)_*\right) + \left(\left(\left(\ell \cdot n\right) \cdot \left(U \cdot 2\right)\right) \cdot \frac{-2}{\frac{Om}{\ell}}\right))_*}\\ \end{array}\]

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 3 regimes

  2. if (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 4.2799990785414e-313

    1. Initial program 50.7

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

    3. Applied add-cube-cbrt50.7

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

    4. Applied prod-diff50.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left((\left(\sqrt[3]{t - 2 \cdot \frac{\ell \cdot \ell}{Om}} \cdot \sqrt[3]{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right) \cdot \left(\sqrt[3]{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right) + \left(-\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right))_* + (\left(-\left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + \left(\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right))_*\right)}}\]

    5. Applied distribute-rgt-in50.7

      \[\leadsto \sqrt{\color{blue}{(\left(\sqrt[3]{t - 2 \cdot \frac{\ell \cdot \ell}{Om}} \cdot \sqrt[3]{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right) \cdot \left(\sqrt[3]{t - 2 \cdot \frac{\ell \cdot \ell}{Om}}\right) + \left(-\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right))_* \cdot \left(\left(2 \cdot n\right) \cdot U\right) + (\left(-\left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + \left(\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right))_* \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\]

    6. Applied simplify48.9

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

    7. Applied simplify47.6

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

    if 4.2799990785414e-313 < (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 3.4777793799200777e+248

    1. Initial program 8.2

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

    3. Applied associate-*l*1.1

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

    if 3.4777793799200777e+248 < (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

    1. Initial program 52.4

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

    3. Applied add-sqr-sqrt56.7

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

    4. Applied prod-diff56.7

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

    5. Applied distribute-rgt-in56.7

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

    6. Applied simplify47.1

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

    7. Applied simplify46.7

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

    9. Applied *-un-lft-identity46.7

      \[\leadsto \sqrt{\color{blue}{1 \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot \left(-2\right)\right) + \left((\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U* - U\right)\right) + t)_*\right))_* + \left(n \cdot 2\right) \cdot \left(U \cdot 0\right)\right)}}\]

    10. Applied sqrt-prod46.7

      \[\leadsto \color{blue}{\sqrt{1} \cdot \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\ell \cdot \left(-2\right)\right) + \left((\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U* - U\right)\right) + t)_*\right))_* + \left(n \cdot 2\right) \cdot \left(U \cdot 0\right)}}\]

    11. Applied simplify44.7

      \[\leadsto \sqrt{1} \cdot \color{blue}{\sqrt{(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left((\left(n \cdot \left(U* - U\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) + t)_*\right) + \left(\left(\left(\ell \cdot n\right) \cdot \left(U \cdot 2\right)\right) \cdot \frac{-2}{\frac{Om}{\ell}}\right))_*}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed 2019053 +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*))))))