Average Error: 33.8 → 24.1
Time: 4.5m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \le 6.8526928380684 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)}\\ \mathbf{if}\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)} \le 2.889543868744176 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t \cdot \left(\left(2 \cdot n\right) \cdot U\right) + \left(\left(n \cdot \left(2 \cdot U\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\left(-2 \cdot \ell\right) - \left(n \cdot \left(U - U*\right)\right) \cdot \frac{\ell}{Om}\right)}\\ \end{array}\]

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (sqrt (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))) < 6.8526928380684e-310

    1. Initial program 49.2

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

    3. Applied add-sqr-sqrt49.2

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

    4. Taylor expanded around 0 49.2

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

    5. Applied simplify47.2

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

    7. Applied sqrt-prod44.5

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

    if 6.8526928380684e-310 < (sqrt (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))) < 2.889543868744176e+128

    1. Initial program 8.7

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

    3. Applied add-sqr-sqrt9.0

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

    4. Taylor expanded around 0 9.0

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

    5. Applied simplify8.4

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

    7. Applied associate-*l*1.4

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

    if 2.889543868744176e+128 < (sqrt (* (* 2 n) (* U (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))

    1. Initial program 53.2

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

    3. Applied add-sqr-sqrt53.2

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

    4. Taylor expanded around 0 53.2

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

    5. Applied simplify45.8

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

    7. Applied sub-neg45.8

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

    8. Applied associate--l+45.8

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

    9. Applied distribute-rgt-in45.8

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

    10. Applied simplify39.8

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

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed 2019053 
(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*))))))