Average Error: 33.0 → 22.9
Time: 2.6m
Precision: 64
Internal Precision: 576
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left((n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) + \left(\left(t \cdot 2\right) \cdot \left(n \cdot U\right)\right))_* \le 1.0770963626450554 \cdot 10^{-301}:\\ \;\;\;\;\left|{\left((\left((n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\left(n \cdot \left(-2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)\right) + \left(n \cdot \left(\left(t \cdot 2\right) \cdot U\right)\right))_*\right)}^{\frac{1}{2}}\right|\\ \mathbf{elif}\;(\left((n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) + \left(\left(t \cdot 2\right) \cdot \left(n \cdot U\right)\right))_* \le 2.0685079534963795 \cdot 10^{+307}:\\ \;\;\;\;\left|\sqrt{(\left((n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) + \left(\left(t \cdot 2\right) \cdot \left(n \cdot U\right)\right))_*}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(\left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_*} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array}\]

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 (fma (fma n (* (- U U*) (/ l Om)) (* l 2)) (* (/ l Om) (* (- U) (* n 2))) (* (* t 2) (* U n))) < 1.0770963626450554e-301

    1. Initial program 55.4

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

    2. Initial simplification55.0

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

    4. Applied sub-neg55.0

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

    5. Applied distribute-rgt-in55.0

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

    6. Simplified54.7

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

    8. Applied add-sqr-sqrt54.7

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

    9. Applied rem-sqrt-square54.7

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

    10. Simplified49.8

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

    12. Applied associate-*r*40.5

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

    14. Applied pow1/240.5

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

    if 1.0770963626450554e-301 < (fma (fma n (* (- U U*) (/ l Om)) (* l 2)) (* (/ l Om) (* (- U) (* n 2))) (* (* t 2) (* U n))) < 2.0685079534963795e+307

    1. Initial program 11.1

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

    2. Initial simplification8.5

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

    4. Applied sub-neg8.5

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

    5. Applied distribute-rgt-in8.5

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

    6. Simplified4.5

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

    8. Applied add-sqr-sqrt4.5

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

    9. Applied rem-sqrt-square4.5

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

    10. Simplified1.9

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

    12. Applied associate-*l*0.9

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

    if 2.0685079534963795e+307 < (fma (fma n (* (- U U*) (/ l Om)) (* l 2)) (* (/ l Om) (* (- U) (* n 2))) (* (* t 2) (* U n)))

    1. Initial program 58.8

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

    2. Initial simplification59.2

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

    4. Applied sqrt-prod52.9

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

  4. Final simplification22.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;(\left((n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) + \left(\left(t \cdot 2\right) \cdot \left(n \cdot U\right)\right))_* \le 1.0770963626450554 \cdot 10^{-301}:\\ \;\;\;\;\left|{\left((\left((n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\left(n \cdot \left(-2\right)\right) \cdot \left(\frac{\ell}{Om} \cdot U\right)\right) + \left(n \cdot \left(\left(t \cdot 2\right) \cdot U\right)\right))_*\right)}^{\frac{1}{2}}\right|\\ \mathbf{elif}\;(\left((n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) + \left(\left(t \cdot 2\right) \cdot \left(n \cdot U\right)\right))_* \le 2.0685079534963795 \cdot 10^{+307}:\\ \;\;\;\;\left|\sqrt{(\left((n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) + \left(\ell \cdot 2\right))_*\right) \cdot \left(\left(U \cdot \left(n \cdot 2\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) + \left(\left(t \cdot 2\right) \cdot \left(n \cdot U\right)\right))_*}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - (\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U - U*\right) \cdot n\right) + \left(\left(\ell \cdot 2\right) \cdot \frac{\ell}{Om}\right))_*} \cdot \sqrt{2 \cdot \left(n \cdot U\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018225 +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*))))))