Average Error: 33.1 → 29.5
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -499.91336675588843:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;n \le -5.619775295235107 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\\ \mathbf{elif}\;n \le 2.442827692742167 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\frac{U}{Om} - \frac{U*}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot n\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

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 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 < -5.619775295235107e-267

    1. Initial program 33.8

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

    3. Applied associate-/l*31.1

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

    5. Applied add-sqr-sqrt31.3

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

    if -5.619775295235107e-267 < n < 2.442827692742167e-300

    1. Initial program 40.0

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

    3. Applied associate-/l*37.0

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

    5. Applied associate-*l*36.9

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

    6. Taylor expanded around 0 43.6

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

    7. Simplified38.0

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

    if 2.442827692742167e-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 simplification29.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -499.91336675588843:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;n \le -5.619775295235107 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\\ \mathbf{elif}\;n \le 2.442827692742167 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(\frac{U}{Om} - \frac{U*}{Om}\right) \cdot \left(\frac{\ell}{Om} \cdot \left(\ell \cdot n\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t - \frac{\ell}{\frac{Om}{\ell}} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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

sample375.0ms

Algorithm
intervals

simplify94.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

94.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

prune31.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 32.6b

localize62.0ms

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*))

rewrite90.0ms

Algorithm
rewrite-expression-head
Counts
4 → 76
Calls

4 calls. Slowest were:

49.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
21.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
12.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))

series1.7s

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*)))))
338.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
115.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))
16.0ms
(/ (* l l) Om)

simplify7.4s

Counts
52 → 88
Calls

52 calls. Slowest were:

336.0ms
(* (+ t (* 2 (/ (* l l) Om))) (+ U U*))
335.0ms
(sqrt (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))
316.0ms
(cbrt (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

prune1.8s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 22.4b

localize18.0ms

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*)))

rewrite90.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

41.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
20.0ms
(sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
14.0ms
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))

series1.7s

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*)))))
248.0ms
(* (* (* 2 n) U) (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
207.0ms
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))
109.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))

simplify10.9s

Counts
65 → 100
Calls

65 calls. Slowest were:

641.0ms
(+ (log (* (* 2 n) U)) (log (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))))
331.0ms
(* (- U*) (* n (pow (/ l Om) 2)))
329.0ms
(sqrt (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))

prune2.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 21.9b

localize10.0ms

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*)))

rewrite60.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

24.0ms
(* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
16.0ms
(sqrt (* (* 2 n) (* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))))
12.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))

series1.7s

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*))))))
252.0ms
(* U (- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*))))
196.0ms
(- (- t (* 2 (/ l (/ Om l)))) (* (* n (pow (/ l Om) 2)) (- U U*)))
102.0ms
(* (* n (pow (/ l Om) 2)) (- U U*))

simplify13.4s

Counts
61 → 96
Calls

61 calls. Slowest were:

753.0ms
(* (- (* (* n (pow (/ l Om) 2)) (- U U*))) U)
512.0ms
(* U (- (* (* n (pow (/ l Om) 2)) (- U U*))))
426.0ms
(* U (- (* (* n (pow (/ l Om) 2)) (- U U*))))

prune1.9s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 17.5b

localize22.0ms

Local error

Found 4 expressions with local error:

32.3b
(cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))
32.3b
(cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))
32.3b
(cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))
19.2b
(sqrt (* (* 2 n) (* U (- (- t (* 2 (/ l (/ Om l)))) (* (* (cbrt (* (* n (pow (/ l Om) 2)) (- U U*))) (cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))) (cbrt (* (* n (pow (/ l Om) 2)) (- U U*))))))))

rewrite301.0ms

Algorithm
rewrite-expression-head
Counts
4 → 96
Calls

4 calls. Slowest were:

282.0ms
(sqrt (* (* 2 n) (* U (- (- t (* 2 (/ l (/ Om l)))) (* (* (cbrt (* (* n (pow (/ l Om) 2)) (- U U*))) (cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))) (cbrt (* (* n (pow (/ l Om) 2)) (- U U*))))))))
4.0ms
(cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))
4.0ms
(cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))

series1.9s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.2s
(sqrt (* (* 2 n) (* U (- (- t (* 2 (/ l (/ Om l)))) (* (* (cbrt (* (* n (pow (/ l Om) 2)) (- U U*))) (cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))) (cbrt (* (* n (pow (/ l Om) 2)) (- U U*))))))))
269.0ms
(cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))
202.0ms
(cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))
192.0ms
(cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))

simplify18.2s

Counts
76 → 108
Calls

76 calls. Slowest were:

597.0ms
(sqrt (* (* 2 n) (* U (- (* (- (pow t 3) (pow (* 2 (/ l (/ Om l))) 3)) (cbrt (+ U U*))) (* (+ (* t t) (+ (* (* 2 (/ l (/ Om l))) (* 2 (/ l (/ Om l)))) (* t (* 2 (/ l (/ Om l)))))) (* (* (cbrt (* (* n (pow (/ l Om) 2)) (- (* U U) (* U* U*)))) (cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))) (cbrt (* (* n (pow (/ l Om) 2)) (- U U*)))))))))
427.0ms
(* (cbrt -1) (exp (* 1/3 (- (+ (* 2 (log (/ -1 Om))) (log -1)) (+ (* 2 (log (/ -1 l))) (+ (log (/ -1 n)) (log (/ -1 U))))))))
408.0ms
(* (cbrt -1) (exp (* 1/3 (- (+ (* 2 (log (/ -1 Om))) (log -1)) (+ (* 2 (log (/ -1 l))) (+ (log (/ -1 n)) (log (/ -1 U))))))))

prune3.3s

Pruning

18 alts after pruning (17 fresh and 1 done)

Merged error: 17.5b

regimes1.3s

Accuracy

6% (11.9b remaining)

Error of 29.5b against oracle of 17.7b and baseline of 30.3b

bsearch1.4s

end0.0ms

sample7.4s

Algorithm
intervals