Average Error: 32.8 → 27.8
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.044868122882883 \cdot 10^{-308}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{n \cdot \ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \frac{1}{Om}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \mathbf{elif}\;t \le 3.1666364102152156 \cdot 10^{-51} \lor \neg \left(t \le 4.255718500878035 \cdot 10^{+206}\right):\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot 2} \cdot \sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{n \cdot \ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{n \cdot \ell}{\frac{Om}{-2 \cdot U}}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \end{array}\]

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 t < -5.044868122882883e-308

    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. Initial simplification31.5

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

    4. Applied sub-neg31.5

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

    5. Applied distribute-rgt-in31.5

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

    6. Simplified28.6

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

    8. Applied add-sqr-sqrt28.6

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

    9. Applied rem-sqrt-square28.6

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

    10. Simplified27.7

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

    12. Applied associate-*l/27.0

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

    14. Applied div-inv27.0

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

    if -5.044868122882883e-308 < t < 3.1666364102152156e-51 or 4.255718500878035e+206 < t

    1. Initial program 35.7

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

    2. Initial simplification34.3

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

    4. Applied sqrt-prod31.5

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

    if 3.1666364102152156e-51 < t < 4.255718500878035e+206

    1. Initial program 29.7

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

    2. Initial simplification28.8

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

    4. Applied sub-neg28.8

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

    5. Applied distribute-rgt-in28.8

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

    6. Simplified26.3

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

    8. Applied add-sqr-sqrt26.3

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

    9. Applied rem-sqrt-square26.3

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

    10. Simplified25.3

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

    12. Applied associate-*l/24.5

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

    14. Applied associate-/l*24.9

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

  4. Final simplification27.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.044868122882883 \cdot 10^{-308}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{n \cdot \ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\left(\left(-2 \cdot U\right) \cdot \left(n \cdot \ell\right)\right) \cdot \frac{1}{Om}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \mathbf{elif}\;t \le 3.1666364102152156 \cdot 10^{-51} \lor \neg \left(t \le 4.255718500878035 \cdot 10^{+206}\right):\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot 2} \cdot \sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \left(U - U*\right)\right) + \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{n \cdot \ell}{Om}\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{n \cdot \ell}{\frac{Om}{-2 \cdot U}}\right) + \left(\left(U \cdot t\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018258 +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*))))))