Average Error: 33.1 → 26.4
Time: 9.8m
Precision: 64
Internal Precision: 384
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(n \cdot 2\right) \cdot U \le -1.5946008831930408 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(-2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;\left(n \cdot 2\right) \cdot U \le 3.0906770541245 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot \left(U* \cdot U\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{n}{Om}\right)\right) \cdot 2 - \frac{4 \cdot n}{Om} \cdot \left(\left(\ell \cdot \ell\right) \cdot U\right)\right) - 2 \cdot \left(\left(\left(\ell \cdot U\right) \cdot \left(\ell \cdot U\right)\right) \cdot \left(\frac{n}{Om} \cdot \frac{n}{Om}\right) - \left(U \cdot t\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

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 (* (* n 2) U) < -1.5946008831930408e-149

    1. Initial program 25.9

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

    3. Applied sub-neg25.9

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

    4. Applied associate--l+25.9

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

    5. Applied distribute-lft-in25.9

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

    if -1.5946008831930408e-149 < (* (* n 2) U) < 3.0906770541245e-319

    1. Initial program 47.1

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

    3. Applied sub-neg47.1

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

    4. Applied associate--l+47.1

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

    5. Applied distribute-lft-in47.1

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

    6. Taylor expanded around inf 47.2

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

    7. Applied simplify35.6

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

    if 3.0906770541245e-319 < (* (* n 2) U)

    1. Initial program 28.4

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

    3. Applied sqrt-prod20.4

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

Runtime

Time bar (total: 9.8m)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(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*))))))