Average Error: 33.0 → 26.4
Time: 41.9s
Precision: 64
Internal Precision: 128
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -1.2052424836324358 \cdot 10^{+75} \lor \neg \left(U \le 1.0594215529279111 \cdot 10^{-16}\right):\\ \;\;\;\;\sqrt{\sqrt{\left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* + t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* + t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(-2 \cdot U\right) \cdot \left(\ell \cdot n\right)}{Om}\right) + \left(\left(t \cdot U\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \end{array}\]

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 2 regimes

  2. if U < -1.2052424836324358e+75 or 1.0594215529279111e-16 < U

    1. Initial program 27.8

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

    2. Initial simplification25.3

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

    4. Applied sub-neg25.3

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

    5. Applied distribute-rgt-in25.3

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

    6. Simplified23.2

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

    8. Applied add-sqr-sqrt23.4

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

    if -1.2052424836324358e+75 < U < 1.0594215529279111e-16

    1. Initial program 35.6

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

    2. Initial simplification35.2

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

    4. Applied sub-neg35.2

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

    5. Applied distribute-rgt-in35.2

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

    6. Simplified31.5

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

    8. Applied add-sqr-sqrt31.5

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

    9. Applied rem-sqrt-square31.5

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

    10. Simplified27.8

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -1.2052424836324358 \cdot 10^{+75} \lor \neg \left(U \le 1.0594215529279111 \cdot 10^{-16}\right):\\ \;\;\;\;\sqrt{\sqrt{\left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* + t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(\frac{\ell}{Om} \cdot \left(n \cdot \left(-2 \cdot U\right)\right)\right) \cdot (\left(n \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om}\right) + \left(2 \cdot \ell\right))_* + t \cdot \left(\left(n \cdot U\right) \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{(\left((\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right) + \left(2 \cdot \ell\right))_*\right) \cdot \left(\frac{\left(-2 \cdot U\right) \cdot \left(\ell \cdot n\right)}{Om}\right) + \left(\left(t \cdot U\right) \cdot \left(2 \cdot n\right)\right))_*}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +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: 27.9s)Debug log

start357.0ms

Algorithm
intervals

setup143.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 28.4b

localize44.0ms

Local error

Found 4 expressions with local error:

15.7b
(sqrt (* (* 2 (* U n)) (- t (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l))))))
13.7b
(fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l)))
10.0b
(* (* 2 (* U n)) (- t (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l)))))
0.2b
(* (/ l Om) (/ l Om))

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 73
Calls

4 calls. Slowest were:

5.0ms
(* (* 2 (* U n)) (- t (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l)))))
4.0ms
(sqrt (* (* 2 (* U n)) (- t (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l))))))
3.0ms
(* (/ l Om) (/ l Om))

series1.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.4s
(sqrt (* (* 2 (* U n)) (- t (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l))))))
238.0ms
(* (* 2 (* U n)) (- t (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l)))))
107.0ms
(fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l)))
20.0ms
(* (/ l Om) (/ l Om))

simplify5.5s

Counts
42 → 85
Calls

42 calls. Slowest were:

941.0ms
(* (* (* (/ l Om) (/ l Om)) (/ l Om)) (* (* (/ l Om) (/ l Om)) (/ l Om)))
704.0ms
(- (+ (* 2 (/ (pow l 2) Om)) (/ (* U (* n (pow l 2))) (pow Om 2))) (/ (* n (* U* (pow l 2))) (pow Om 2)))
587.0ms
(+ (* t t) (+ (* (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l))) (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l)))) (* t (fma (* (/ l Om) (/ l Om)) (* (- U U*) n) (* (/ l Om) (* 2 l))))))

prune1.4s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 16.3b

localize33.0ms

Local error

Found 4 expressions with local error:

15.7b
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
7.6b
(fma (* n (- U U*)) (/ l Om) (* 2 l))
7.5b
(* (* n (* U -2)) (/ l Om))
3.5b
(* t (* 2 (* U n)))

rewrite20.0ms

Algorithm
rewrite-expression-head
Counts
4 → 56
Calls

4 calls. Slowest were:

9.0ms
(* (* n (* U -2)) (/ l Om))
6.0ms
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
4.0ms
(* t (* 2 (* U n)))

series870.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

727.0ms
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
81.0ms
(fma (* n (- U U*)) (/ l Om) (* 2 l))
38.0ms
(* (* n (* U -2)) (/ l Om))
25.0ms
(* t (* 2 (* U n)))

simplify2.9s

Counts
26 → 68
Calls

26 calls. Slowest were:

317.0ms
(+ (log (* n (* U -2))) (log (/ l Om)))
311.0ms
(* -2 (/ (* U (* n l)) Om))
264.0ms
(sqrt (- (* (* t (* 2 (* U n))) (* t (* 2 (* U n)))) (* (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l))))))

prune1.4s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 14.8b

localize15.0ms

Local error

Found 4 expressions with local error:

15.7b
(sqrt (+ (* t (* 2 (* U n))) (* (* n (* (* U -2) (/ l Om))) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
7.6b
(fma (* n (- U U*)) (/ l Om) (* 2 l))
6.5b
(* n (* (* U -2) (/ l Om)))
4.6b
(* (* U -2) (/ l Om))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

7.0ms
(* n (* (* U -2) (/ l Om)))
7.0ms
(* (* U -2) (/ l Om))
4.0ms
(sqrt (+ (* t (* 2 (* U n))) (* (* n (* (* U -2) (/ l Om))) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))

series921.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

763.0ms
(sqrt (+ (* t (* 2 (* U n))) (* (* n (* (* U -2) (/ l Om))) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
81.0ms
(fma (* n (- U U*)) (/ l Om) (* 2 l))
48.0ms
(* n (* (* U -2) (/ l Om)))
29.0ms
(* (* U -2) (/ l Om))

simplify2.8s

Counts
26 → 69
Calls

26 calls. Slowest were:

521.0ms
(* (* (* (* U -2) (* U -2)) (* U -2)) (* (* (/ l Om) (/ l Om)) (/ l Om)))
247.0ms
(sqrt (+ (pow (* t (* 2 (* U n))) 3) (pow (* (* n (* (* U -2) (/ l Om))) (fma (* n (- U U*)) (/ l Om) (* 2 l))) 3)))
245.0ms
(* -2 (/ (* U (* n l)) Om))

prune1.6s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 14.8b

localize8.0ms

Local error

Found 4 expressions with local error:

15.7b
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
15.7b
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
7.6b
(fma (* n (- U U*)) (/ l Om) (* 2 l))
7.6b
(fma (* n (- U U*)) (/ l Om) (* 2 l))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

3.0ms
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
3.0ms
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
0.0ms
(fma (* n (- U U*)) (/ l Om) (* 2 l))

series1.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

804.0ms
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
734.0ms
(sqrt (+ (* t (* 2 (* U n))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l)))))
70.0ms
(fma (* n (- U U*)) (/ l Om) (* 2 l))
68.0ms
(fma (* n (- U U*)) (/ l Om) (* 2 l))

simplify2.6s

Counts
24 → 64
Calls

24 calls. Slowest were:

272.0ms
(sqrt (- (* (* t (* 2 (* U n))) (* t (* 2 (* U n)))) (* (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l))) (* (* (* n (* U -2)) (/ l Om)) (fma (* n (- U U*)) (/ l Om) (* 2 l))))))
227.0ms
(- (+ (/ (* U (* n l)) Om) (* 2 l)) (/ (* n (* U* l)) Om))
213.0ms
(- (+ (/ (* U (* n l)) Om) (* 2 l)) (/ (* n (* U* l)) Om))

prune1.7s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 14.8b

regimes1.2s

Accuracy

24.8% (7.9b remaining)

Error of 26.4b against oracle of 18.5b and baseline of 29.0b

bsearch915.0ms