Average Error: 32.1 → 25.3
Time: 3.1m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \le +\infty:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{(\left((\left(\left(-n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{U - U*}{\frac{Om}{\ell}}\right) + t)_*\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right) + \left(\left(n \cdot \ell\right) \cdot \left(\left(-4 \cdot U\right) \cdot \frac{\ell}{Om}\right)\right))_*}\right| \cdot \sqrt{\sqrt[3]{\sqrt{(\left((\left(\frac{U - U*}{\frac{Om}{\ell}}\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right) + t)_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(-4 \cdot \frac{\ell}{Om}\right)\right))_*}} \cdot \sqrt[3]{\sqrt{(\left((\left(\frac{U - U*}{\frac{Om}{\ell}}\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right) + t)_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(-4 \cdot \frac{\ell}{Om}\right)\right))_*}}}\\ \end{array}\]

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 (* (sqrt (* (* 2 n) U)) (sqrt (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < +inf.0

    1. Initial program 27.8

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

    3. Applied sqrt-prod19.9

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

    if +inf.0 < (* (sqrt (* (* 2 n) U)) (sqrt (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

    1. Initial program 35.4

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

    3. Applied *-un-lft-identity35.4

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

    4. Applied prod-diff35.4

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

    5. Applied distribute-rgt-in35.4

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

    6. Applied simplify32.2

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

    7. Applied simplify31.4

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

    9. Applied fma-udef31.4

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

    10. Applied associate--l+31.4

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

    11. Applied distribute-rgt-in31.4

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

    12. Applied simplify28.7

      \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot -4\right)} + \left(t - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) + n \cdot \left(U \cdot 0\right)}\]
    13. Using strategy rm

    14. Applied add-cube-cbrt29.1

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot -4\right) + \left(t - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) + n \cdot \left(U \cdot 0\right)} \cdot \sqrt[3]{\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot -4\right) + \left(t - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) + n \cdot \left(U \cdot 0\right)}\right) \cdot \sqrt[3]{\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot -4\right) + \left(t - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) + n \cdot \left(U \cdot 0\right)}}}\]

    15. Applied sqrt-prod29.1

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot -4\right) + \left(t - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) + n \cdot \left(U \cdot 0\right)} \cdot \sqrt[3]{\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot -4\right) + \left(t - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) + n \cdot \left(U \cdot 0\right)}} \cdot \sqrt{\sqrt[3]{\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot -4\right) + \left(t - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) + n \cdot \left(U \cdot 0\right)}}}\]

    16. Applied simplify29.3

      \[\leadsto \color{blue}{\left|\sqrt[3]{(\left((\left(\left(-n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{U - U*}{\frac{Om}{\ell}}\right) + t)_*\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right) + \left(\left(n \cdot \ell\right) \cdot \left(\left(-4 \cdot U\right) \cdot \frac{\ell}{Om}\right)\right))_*}\right|} \cdot \sqrt{\sqrt[3]{\left(\left(\frac{\ell}{Om} \cdot \left(n \cdot \ell\right)\right) \cdot \left(U \cdot -4\right) + \left(t - \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(\left(U \cdot n\right) \cdot 2\right)\right) + n \cdot \left(U \cdot 0\right)}}\]

    17. Applied simplify29.3

      \[\leadsto \left|\sqrt[3]{(\left((\left(\left(-n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{U - U*}{\frac{Om}{\ell}}\right) + t)_*\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right) + \left(\left(n \cdot \ell\right) \cdot \left(\left(-4 \cdot U\right) \cdot \frac{\ell}{Om}\right)\right))_*}\right| \cdot \color{blue}{\sqrt{\sqrt[3]{(\left((\left(\frac{U - U*}{\frac{Om}{\ell}}\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right) + t)_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(-4 \cdot \frac{\ell}{Om}\right)\right))_*}}}\]
    18. Using strategy rm

    19. Applied add-sqr-sqrt29.3

      \[\leadsto \left|\sqrt[3]{(\left((\left(\left(-n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{U - U*}{\frac{Om}{\ell}}\right) + t)_*\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right) + \left(\left(n \cdot \ell\right) \cdot \left(\left(-4 \cdot U\right) \cdot \frac{\ell}{Om}\right)\right))_*}\right| \cdot \sqrt{\sqrt[3]{\color{blue}{\sqrt{(\left((\left(\frac{U - U*}{\frac{Om}{\ell}}\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right) + t)_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(-4 \cdot \frac{\ell}{Om}\right)\right))_*} \cdot \sqrt{(\left((\left(\frac{U - U*}{\frac{Om}{\ell}}\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right) + t)_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(-4 \cdot \frac{\ell}{Om}\right)\right))_*}}}}\]

    20. Applied cbrt-prod29.3

      \[\leadsto \left|\sqrt[3]{(\left((\left(\left(-n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{U - U*}{\frac{Om}{\ell}}\right) + t)_*\right) \cdot \left(\left(n \cdot 2\right) \cdot U\right) + \left(\left(n \cdot \ell\right) \cdot \left(\left(-4 \cdot U\right) \cdot \frac{\ell}{Om}\right)\right))_*}\right| \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt{(\left((\left(\frac{U - U*}{\frac{Om}{\ell}}\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right) + t)_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(-4 \cdot \frac{\ell}{Om}\right)\right))_*}} \cdot \sqrt[3]{\sqrt{(\left((\left(\frac{U - U*}{\frac{Om}{\ell}}\right) \cdot \left(\left(-n\right) \cdot \frac{\ell}{Om}\right) + t)_*\right) \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(\left(\left(n \cdot \ell\right) \cdot U\right) \cdot \left(-4 \cdot \frac{\ell}{Om}\right)\right))_*}}}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +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*))))))