Average Error: 33.2 → 25.6
Time: 3.1m
Precision: 64
Internal Precision: 384
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(U \cdot 2\right) \cdot n \le -3.0632070042157 \cdot 10^{-322}:\\ \;\;\;\;{\left(\left(\left(U \cdot n\right) \cdot 2\right) \cdot (\left(\frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) + \left(t - \left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_* + \left(0 \cdot n\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{if}\;\left(U \cdot 2\right) \cdot n \le 0.0:\\ \;\;\;\;\sqrt{(2 \cdot \left(\left(\frac{n}{Om} \cdot \frac{n}{Om}\right) \cdot \left(\left(U* \cdot \ell\right) \cdot \left(\ell \cdot U\right) - \left(\ell \cdot U\right) \cdot \left(\ell \cdot U\right)\right)\right) + \left(\left(t \cdot U\right) \cdot \left(n \cdot 2\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

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 (* (* U 2) n) < -3.0632070042157e-322

    1. Initial program 29.2

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

    3. Applied add-sqr-sqrt59.5

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

    4. Applied prod-diff59.5

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

    5. Applied distribute-rgt-in59.5

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

    6. Applied simplify25.9

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

    7. Applied simplify25.0

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

    9. Applied pow1/225.0

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

    if -3.0632070042157e-322 < (* (* U 2) n) < 0.0

    1. Initial program 57.1

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

    3. Applied add-sqr-sqrt59.5

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

    4. Applied prod-diff59.5

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

    5. Applied distribute-rgt-in59.5

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

    6. Applied simplify57.0

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

    7. Applied simplify57.0

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

    9. Applied pow1/257.0

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

    10. Taylor expanded around inf 46.8

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

    11. Applied simplify41.3

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

    if 0.0 < (* (* U 2) n)

    1. Initial program 27.7

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

    3. Applied sqrt-prod19.9

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

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +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*))))))