Average Error: 33.6 → 28.8
Time: 3.4m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.4567198136425585 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \cdot \sqrt{\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \mathbf{elif}\;t \le 2.122126985655257 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

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 t < 1.4567198136425585e-296

    1. Initial program 34.0

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

    3. Applied associate-/l*31.3

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

    4. Taylor expanded around inf 37.3

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

    5. Simplified30.4

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

    7. Applied associate-*l/30.1

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

    9. Applied add-sqr-sqrt30.3

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

    if 1.4567198136425585e-296 < t < 2.122126985655257e+191

    1. Initial program 32.5

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

    3. Applied associate-/l*29.9

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

    4. Taylor expanded around inf 36.3

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

    5. Simplified29.0

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

    7. Applied associate-*l/28.7

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

    9. Applied associate-*l*28.5

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

    if 2.122126985655257e+191 < t

    1. Initial program 36.3

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

    3. Applied associate-/l*34.4

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

    4. Taylor expanded around inf 40.1

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

    5. Simplified34.2

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

    7. Applied sqrt-prod22.5

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

  4. Final simplification28.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.4567198136425585 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \cdot \sqrt{\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\\ \mathbf{elif}\;t \le 2.122126985655257 \cdot 10^{+191}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +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*))))))