Average Error: 33.1 → 28.6
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -499.91336675588843:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;n \le -4.306424586034209 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left((\left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_*\right))_* \cdot U\right)}\\ \mathbf{elif}\;n \le 3.615880178248846 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

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 n < -499.91336675588843

    1. Initial program 30.6

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

    3. Applied associate-/l*28.4

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

    5. Applied sqrt-prod39.6

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

    if -499.91336675588843 < n < -4.306424586034209e-281

    1. Initial program 34.3

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

    3. Applied associate-/l*31.6

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

    5. Applied associate-*l*29.6

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

    7. Applied add-cube-cbrt29.9

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

    8. Applied prod-diff29.9

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

    9. Simplified29.6

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

    10. Simplified28.6

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

    if -4.306424586034209e-281 < n < 3.615880178248846e-300

    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. Using strategy rm

    3. Applied associate-/l*36.4

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

    5. Applied associate-*l*36.3

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

    7. Applied add-cube-cbrt36.5

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

    8. Applied prod-diff36.5

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

    9. Simplified36.3

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

    10. Simplified34.4

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

    12. Applied pow134.4

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

    13. Applied pow134.4

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

    14. Applied pow-prod-down34.4

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

    15. Simplified37.0

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

    if 3.615880178248846e-300 < n

    1. Initial program 33.0

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

    3. Applied associate-/l*30.2

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

    5. Applied associate-*l*30.4

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

    7. Applied sqrt-prod23.5

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification28.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -499.91336675588843:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;n \le -4.306424586034209 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left((\left(\frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left((-2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}}\right) + t)_*\right))_* \cdot U\right)}\\ \mathbf{elif}\;n \le 3.615880178248846 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot (\left(n \cdot \left(\left(U* - U\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left((\left(\ell \cdot \frac{\ell}{Om}\right) \cdot -2 + t)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +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*))))))

Details

Time bar (total: 1.3m)Debug log

sample351.0ms

Algorithm
intervals

simplify113.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

113.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.6b

localize57.0ms

Local error

Found 4 expressions with local error:

19.2b
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
9.6b
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
6.1b
(/ (* l l) Om)
6.0b
(* (* n (pow (/ l Om) 2)) (- U U*))

rewrite95.0ms

Algorithm
rewrite-expression-head
Counts
4 → 90
Calls

4 calls. Slowest were:

43.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
26.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
13.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))

series1.6s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.2s
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
359.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
89.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
14.0ms
(/ (* l l) Om)

simplify8.3s

Counts
58 → 102
Calls

58 calls. Slowest were:

621.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
526.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
495.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))

prune1.4s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 22.6b

localize20.0ms

Local error

Found 4 expressions with local error:

19.2b
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
9.6b
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
6.0b
(* (* n (pow (/ l Om) 2)) (- U U*))
4.8b
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))

rewrite65.0ms

Algorithm
rewrite-expression-head
Counts
4 → 108
Calls

4 calls. Slowest were:

28.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
12.0ms
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))
12.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

series1.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.1s
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
331.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
246.0ms
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))
68.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))

simplify15.3s

Counts
79 → 120
Calls

79 calls. Slowest were:

668.0ms
(fma 1 (- t (* 2 (/ l (/ Om l)))) (- (* (- U U*) (* n (pow (/ l Om) 2)))))
641.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))
539.0ms
(* (fma (- (- U U*)) (* n (pow (/ l Om) 2)) (* (- U U*) (* n (pow (/ l Om) 2)))) (* (* 2 n) U))

prune2.6s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 22.1b

localize9.0ms

Local error

Found 4 expressions with local error:

19.2b
(sqrt (* (* 2 n) (* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))))
6.0b
(* (* n (pow (/ l Om) 2)) (- U U*))
5.0b
(* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
4.8b
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))

rewrite78.0ms

Algorithm
rewrite-expression-head
Counts
4 → 104
Calls

4 calls. Slowest were:

29.0ms
(* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
22.0ms
(sqrt (* (* 2 n) (* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))))
14.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))

series1.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.2s
(sqrt (* (* 2 n) (* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))))
277.0ms
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))
254.0ms
(* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
85.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))

simplify18.7s

Counts
75 → 116
Calls

75 calls. Slowest were:

801.0ms
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))
555.0ms
(* U (- (* (* n (pow (/ l Om) 2)) (- U U*))))
548.0ms
(* U (- (* (* n (pow (/ l Om) 2)) (- U U*))))

prune2.4s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 17.7b

localize34.0ms

Local error

Found 4 expressions with local error:

19.2b
(sqrt (* (* 2 n) (* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0))))
10.7b
(* (/ l Om) (* (- U* U) n))
5.0b
(* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0))
4.6b
(* (* 2 n) (* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0)))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 74
Calls

4 calls. Slowest were:

16.0ms
(* (* 2 n) (* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0)))
14.0ms
(* (/ l Om) (* (- U* U) n))
8.0ms
(* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0))

series2.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.4s
(sqrt (* (* 2 n) (* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0))))
323.0ms
(* (* 2 n) (* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0)))
244.0ms
(* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0))
71.0ms
(* (/ l Om) (* (- U* U) n))

simplify6.0s

Counts
44 → 86
Calls

44 calls. Slowest were:

493.0ms
(sqrt (* (* 2 n) (* U (- (* (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t))) (* 0 0)))))
401.0ms
(* (* (* (* 2 n) (* 2 n)) (* 2 n)) (* (* (* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0)) (* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0))) (* U (+ (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0))))
361.0ms
(+ (* (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t))) (- (* 0 0) (* (fma (* (/ l Om) (* (- U* U) n)) (/ l Om) (fma -2 (/ l (/ Om l)) t)) 0)))

prune1.5s

Pruning

16 alts after pruning (15 fresh and 1 done)

Merged error: 17.5b

regimes1.3s

Accuracy

8.5% (10.9b remaining)

Error of 28.6b against oracle of 17.7b and baseline of 29.6b

bsearch1.8s

end0.0ms

sample9.1s

Algorithm
intervals