Average Error: 32.9 → 27.6
Time: 2.5m
Precision: 64
Internal Precision: 384
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le -2.261217660051935 \cdot 10^{+89}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(\left(\frac{n}{Om} \cdot \ell\right) \cdot U\right) \cdot \left(\ell \cdot 4\right)}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le -1.7000567065397344 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le 2.450450831693368 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \sqrt[3]{{\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right)}^{3}}\right)}\\ \mathbf{if}\;\left(U + U\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell + \ell\right)\right) \le 1.3874370042323594 \cdot 10^{+300}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(n \cdot \left(\frac{U}{Om} \cdot \ell\right)\right) \cdot \left(\ell \cdot 4\right)}\\ \end{array}\]

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 4 regimes

  2. if (* (+ U U) (- t (* (/ l Om) (+ l l)))) < -2.261217660051935e+89

    1. Initial program 39.4

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

    2. Taylor expanded around 0 39.8

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

    3. Applied simplify36.3

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

    5. Applied pow1/236.3

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

    6. Taylor expanded around inf 43.6

      \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) - 4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}}^{\frac{1}{2}}\]

    7. Applied simplify33.6

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

    9. Applied associate-*r*32.9

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

    if -2.261217660051935e+89 < (* (+ U U) (- t (* (/ l Om) (+ l l)))) < -1.7000567065397344e-185 or 2.450450831693368e-165 < (* (+ U U) (- t (* (/ l Om) (+ l l)))) < 1.3874370042323594e+300

    1. Initial program 25.6

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

    3. Applied associate-*l*20.0

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

    if -1.7000567065397344e-185 < (* (+ U U) (- t (* (/ l Om) (+ l l)))) < 2.450450831693368e-165

    1. Initial program 32.0

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

    3. Applied add-cbrt-cube35.1

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

    4. Applied simplify33.3

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

    if 1.3874370042323594e+300 < (* (+ U U) (- t (* (/ l Om) (+ l l))))

    1. Initial program 49.5

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

    2. Taylor expanded around 0 49.4

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

    3. Applied simplify46.7

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

    5. Applied pow1/246.7

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

    6. Taylor expanded around inf 56.7

      \[\leadsto {\color{blue}{\left(2 \cdot \left(n \cdot \left(t \cdot U\right)\right) - 4 \cdot \frac{n \cdot \left({\ell}^{2} \cdot U\right)}{Om}\right)}}^{\frac{1}{2}}\]

    7. Applied simplify39.0

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

    9. Applied div-inv39.0

      \[\leadsto \sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \left(\color{blue}{\left(n \cdot \frac{1}{Om}\right)} \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot 4\right)}\]

    10. Applied associate-*l*37.9

      \[\leadsto \sqrt{\left(t \cdot n\right) \cdot \left(U + U\right) - \color{blue}{\left(n \cdot \left(\frac{1}{Om} \cdot \left(\ell \cdot U\right)\right)\right)} \cdot \left(\ell \cdot 4\right)}\]

    11. Applied simplify37.5

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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*))))))