Average Error: 32.6 → 25.4
Time: 4.4m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{(\left({\left(\sqrt[3]{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\right)}^{3}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right) + \left(\left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right)\right))_* + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \log_* (1 + (e^{0} - 1)^*)} \le 8.011687533618068 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(n \cdot (\left(\frac{\ell}{Om} \cdot 2\right) \cdot \left(-\ell\right) + \left((\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U* - U\right)\right) + t)_*\right))_*\right) + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{if}\;\sqrt{(\left({\left(\sqrt[3]{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\right)}^{3}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right) + \left(\left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right)\right))_* + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \log_* (1 + (e^{0} - 1)^*)} \le 6.853321527358693 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{(\left({\left(\sqrt[3]{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\right)}^{3}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right) + \left(\left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right)\right))_* + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \log_* (1 + (e^{0} - 1)^*)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(-\ell\right) \cdot \left(2 \cdot U\right)\right) \cdot \frac{\ell \cdot n}{\frac{Om}{2}}}\\ \end{array}\]

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 (sqrt (+ (fma (pow (cbrt (- t (* (/ l Om) (* 2 l)))) 3) (* n (* U 2)) (* (* (- n) (- U U*)) (* (* (/ l Om) (/ l Om)) (* n (* U 2))))) (* (* (* 2 n) U) (log1p (expm1 0))))) < 8.011687533618068e-138

    1. Initial program 51.5

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

    3. Applied associate-/l*51.5

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

    5. Applied add-sqr-sqrt57.2

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

    6. Applied prod-diff57.2

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

    7. Applied distribute-rgt-in57.2

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

    8. Applied simplify51.4

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

    9. Applied simplify51.4

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

    11. Applied associate-*l*34.1

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

    if 8.011687533618068e-138 < (sqrt (+ (fma (pow (cbrt (- t (* (/ l Om) (* 2 l)))) 3) (* n (* U 2)) (* (* (- n) (- U U*)) (* (* (/ l Om) (/ l Om)) (* n (* U 2))))) (* (* (* 2 n) U) (log1p (expm1 0))))) < 6.853321527358693e+152

    1. Initial program 7.3

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

    3. Applied add-cube-cbrt7.8

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

    4. Applied prod-diff7.8

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

    5. Applied distribute-lft-in7.8

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

    6. Applied simplify3.0

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

    8. Applied log1p-expm1-u13.8

      \[\leadsto \sqrt{(\left({\left(\sqrt[3]{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\right)}^{3}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right) + \left(\left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right)\right))_* + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\log_* (1 + (e^{(\left(-\left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) + \left(\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right))_*} - 1)^*)}}\]

    9. Applied simplify1.6

      \[\leadsto \sqrt{(\left({\left(\sqrt[3]{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\right)}^{3}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right) + \left(\left(\left(-n\right) \cdot \left(U - U*\right)\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)\right)\right))_* + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \log_* (1 + \color{blue}{(e^{0} - 1)^*})}\]

    if 6.853321527358693e+152 < (sqrt (+ (fma (pow (cbrt (- t (* (/ l Om) (* 2 l)))) 3) (* n (* U 2)) (* (* (- n) (- U U*)) (* (* (/ l Om) (/ l Om)) (* n (* U 2))))) (* (* (* 2 n) U) (log1p (expm1 0)))))

    1. Initial program 55.6

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

    3. Applied associate-/l*55.4

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

    5. Applied add-sqr-sqrt58.0

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

    6. Applied prod-diff58.1

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

    7. Applied distribute-rgt-in58.1

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

    8. Applied simplify60.2

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

    9. Applied simplify60.2

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

    10. Taylor expanded around 0 60.0

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

    11. Applied simplify51.5

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

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed 2018206 +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*))))))