Average Error: 32.2 → 12.1
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.356660349261376 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\ell} \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot t\right)}{\cos k}}\\ \mathbf{elif}\;t \le 2.694066062132053 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right) \cdot t}{\ell} + \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\ell} \cdot 2}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{(\left(\sin k \cdot \frac{k}{t}\right) \cdot k + \left(\left(2 \cdot t\right) \cdot \sin k\right))_* \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\ell}}{\cos k}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -3.356660349261376e-125

    1. Initial program 24.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied unpow324.0

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    4. Applied times-frac17.1

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    5. Applied associate-*l*15.0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied tan-quot15.0

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    8. Applied associate-*l/15.2

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    9. Applied associate-*r/15.3

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    10. Applied frac-times15.9

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \sin k}{\ell \cdot \cos k}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    11. Applied associate-*l/15.7

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\ell \cdot \cos k}}}\]

    12. Simplified10.1

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\sin k \cdot t}{\frac{\ell}{t}} \cdot \left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)}}{\ell \cdot \cos k}}\]
    13. Using strategy rm

    14. Applied times-frac8.5

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\ell} \cdot \frac{\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\cos k}}}\]

    if -3.356660349261376e-125 < t < 2.694066062132053e-108

    1. Initial program 62.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied unpow362.6

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    4. Applied times-frac55.2

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    5. Applied associate-*l*55.2

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied tan-quot55.2

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    8. Applied associate-*l/55.4

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    9. Applied associate-*r/56.6

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    10. Applied frac-times57.4

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \sin k}{\ell \cdot \cos k}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    11. Applied associate-*l/57.4

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\ell \cdot \cos k}}}\]

    12. Simplified43.6

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\sin k \cdot t}{\frac{\ell}{t}} \cdot \left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)}}{\ell \cdot \cos k}}\]

    13. Taylor expanded around inf 24.1

      \[\leadsto \frac{2}{\frac{\color{blue}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\ell} + \frac{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}{\ell}}}{\ell \cdot \cos k}}\]

    if 2.694066062132053e-108 < t

    1. Initial program 23.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    2. Using strategy rm

    3. Applied unpow323.6

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    4. Applied times-frac17.0

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    5. Applied associate-*l*14.8

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied tan-quot14.8

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    8. Applied associate-*l/15.1

      \[\leadsto \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{t \cdot \sin k}{\ell}}\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    9. Applied associate-*r/15.3

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)}{\ell}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    10. Applied frac-times15.6

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \sin k}{\ell \cdot \cos k}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    11. Applied associate-*l/15.4

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(t \cdot \sin k\right)\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\ell \cdot \cos k}}}\]

    12. Simplified9.3

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\sin k \cdot t}{\frac{\ell}{t}} \cdot \left(\left(\sin k \cdot t\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right)}}{\ell \cdot \cos k}}\]

    13. Taylor expanded around inf 9.6

      \[\leadsto \frac{2}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}} \cdot \color{blue}{\left(2 \cdot \left(t \cdot \sin k\right) + \frac{\sin k \cdot {k}^{2}}{t}\right)}}{\ell \cdot \cos k}}\]

    14. Simplified9.0

      \[\leadsto \frac{2}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}} \cdot \color{blue}{(\left(\frac{k}{t} \cdot \sin k\right) \cdot k + \left(\left(2 \cdot t\right) \cdot \sin k\right))_*}}{\ell \cdot \cos k}}\]
    15. Using strategy rm

    16. Applied associate-/r*8.9

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}} \cdot (\left(\frac{k}{t} \cdot \sin k\right) \cdot k + \left(\left(2 \cdot t\right) \cdot \sin k\right))_*}{\ell}}{\cos k}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.356660349261376 \cdot 10^{-125}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sin k \cdot t}{\frac{\ell}{t}}}{\ell} \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot t\right)}{\cos k}}\\ \mathbf{elif}\;t \le 2.694066062132053 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right) \cdot t}{\ell} + \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\ell} \cdot 2}{\cos k \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{(\left(\sin k \cdot \frac{k}{t}\right) \cdot k + \left(\left(2 \cdot t\right) \cdot \sin k\right))_* \cdot \frac{\sin k \cdot t}{\frac{\ell}{t}}}{\ell}}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))

Details

Time bar (total: 1.4m)Debug log

sample421.0ms

Algorithm
intervals

simplify168.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

167.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.0b

localize45.0ms

Local error

Found 4 expressions with local error:

14.9b
(/ (pow t 3) (* l l))
13.6b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
2.9b
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
2.2b
(* (/ (pow t 3) (* l l)) (sin k))

rewrite126.0ms

Algorithm
rewrite-expression-head
Counts
4 → 120
Calls

4 calls. Slowest were:

81.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
25.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
11.0ms
(* (/ (pow t 3) (* l l)) (sin k))

series674.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

364.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
190.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
88.0ms
(* (/ (pow t 3) (* l l)) (sin k))
32.0ms
(/ (pow t 3) (* l l))

simplify15.7s

Counts
98 → 132
Calls

98 calls. Slowest were:

1.0s
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
833.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
681.0ms
(+ (+ (+ (- (log (pow t 3)) (log (* l l))) (log (sin k))) (log (tan k))) (log (+ (+ 1 (pow (/ k t) 2)) 1)))

prune2.3s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 14.7b

localize43.0ms

Local error

Found 4 expressions with local error:

13.6b
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
8.4b
(/ (* t t) l)
3.8b
(* (/ t l) (sin k))
2.9b
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))

rewrite166.0ms

Algorithm
rewrite-expression-head
Counts
4 → 115
Calls

4 calls. Slowest were:

127.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
29.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
4.0ms
(* (/ t l) (sin k))

series727.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

409.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
219.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
88.0ms
(* (/ t l) (sin k))
10.0ms
(/ (* t t) l)

simplify22.3s

Counts
99 → 127
Calls

99 calls. Slowest were:

1.2s
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
978.0ms
(* (* (* l l) (cos k)) (- (+ 1 (pow (/ k t) 2)) 1))
787.0ms
(- (+ 1 (pow (/ k t) 2)) 1)

prune2.0s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 7.4b

localize50.0ms

Local error

Found 4 expressions with local error:

10.9b
(* (/ (* (sin k) t) (/ l t)) (* (* (sin k) t) (fma (/ k t) (/ k t) 2)))
4.0b
(/ (* (/ (* (sin k) t) (/ l t)) (* (* (sin k) t) (fma (/ k t) (/ k t) 2))) (* l (cos k)))
2.4b
(* (* (sin k) t) (fma (/ k t) (/ k t) 2))
1.8b
(/ (* (sin k) t) (/ l t))

rewrite72.0ms

Algorithm
rewrite-expression-head
Counts
4 → 104
Calls

4 calls. Slowest were:

39.0ms
(/ (* (/ (* (sin k) t) (/ l t)) (* (* (sin k) t) (fma (/ k t) (/ k t) 2))) (* l (cos k)))
20.0ms
(* (/ (* (sin k) t) (/ l t)) (* (* (sin k) t) (fma (/ k t) (/ k t) 2)))
7.0ms
(* (* (sin k) t) (fma (/ k t) (/ k t) 2))

series532.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

227.0ms
(/ (* (/ (* (sin k) t) (/ l t)) (* (* (sin k) t) (fma (/ k t) (/ k t) 2))) (* l (cos k)))
164.0ms
(* (/ (* (sin k) t) (/ l t)) (* (* (sin k) t) (fma (/ k t) (/ k t) 2)))
71.0ms
(* (* (sin k) t) (fma (/ k t) (/ k t) 2))
69.0ms
(/ (* (sin k) t) (/ l t))

simplify12.5s

Counts
75 → 116
Calls

75 calls. Slowest were:

1.1s
(/ (* (/ (* (* (* (sin k) t) (* (sin k) t)) (* (sin k) t)) (* (* (/ l t) (/ l t)) (/ l t))) (* (* (* (* (sin k) t) (* (sin k) t)) (* (sin k) t)) (* (* (fma (/ k t) (/ k t) 2) (fma (/ k t) (/ k t) 2)) (fma (/ k t) (/ k t) 2)))) (* (* (* l (cos k)) (* l (cos k))) (* l (cos k))))
937.0ms
(/ (* (* (* (/ (* (sin k) t) (/ l t)) (/ (* (sin k) t) (/ l t))) (/ (* (sin k) t) (/ l t))) (* (* (* (* (sin k) t) (* (sin k) t)) (* (sin k) t)) (* (* (fma (/ k t) (/ k t) 2) (fma (/ k t) (/ k t) 2)) (fma (/ k t) (/ k t) 2)))) (* (* (* l (cos k)) (* l (cos k))) (* l (cos k))))
903.0ms
(/ (* (/ (* (* (* (sin k) t) (* (sin k) t)) (* (sin k) t)) (* (* (/ l t) (/ l t)) (/ l t))) (* (* (* (* (sin k) t) (fma (/ k t) (/ k t) 2)) (* (* (sin k) t) (fma (/ k t) (/ k t) 2))) (* (* (sin k) t) (fma (/ k t) (/ k t) 2)))) (* (* (* l (cos k)) (* l (cos k))) (* l (cos k))))

prune1.8s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 6.0b

localize22.0ms

Local error

Found 4 expressions with local error:

10.9b
(* (/ (* (sin k) t) (/ l t)) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k))))
4.0b
(/ (* (/ (* (sin k) t) (/ l t)) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k)))) (* l (cos k)))
1.8b
(/ (* (sin k) t) (/ l t))
0.4b
(* (* 2 t) (sin k))

rewrite37.0ms

Algorithm
rewrite-expression-head
Counts
4 → 93
Calls

4 calls. Slowest were:

11.0ms
(/ (* (/ (* (sin k) t) (/ l t)) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k)))) (* l (cos k)))
9.0ms
(/ (* (sin k) t) (/ l t))
8.0ms
(* (/ (* (sin k) t) (/ l t)) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k))))

series619.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

249.0ms
(* (/ (* (sin k) t) (/ l t)) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k))))
232.0ms
(/ (* (/ (* (sin k) t) (/ l t)) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k)))) (* l (cos k)))
75.0ms
(/ (* (sin k) t) (/ l t))
62.0ms
(* (* 2 t) (sin k))

simplify7.7s

Counts
63 → 105
Calls

63 calls. Slowest were:

770.0ms
(+ (* 1/6 (/ (* t (pow k 6)) (pow l 2))) (+ (/ (* t (pow k 4)) (pow l 2)) (* 2 (/ (* (pow t 3) (pow k 2)) (pow l 2)))))
667.0ms
(/ (* (/ (* (* (* (sin k) t) (* (sin k) t)) (* (sin k) t)) (* (* (/ l t) (/ l t)) (/ l t))) (* (* (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k))) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k)))) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k))))) (* (* (* l (cos k)) (* l (cos k))) (* l (cos k))))
588.0ms
(/ (* (* (* (/ (* (sin k) t) (/ l t)) (/ (* (sin k) t) (/ l t))) (/ (* (sin k) t) (/ l t))) (* (* (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k))) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k)))) (fma (* (/ k t) (sin k)) k (* (* 2 t) (sin k))))) (* (* (* l (cos k)) (* l (cos k))) (* l (cos k))))

prune1.6s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 5.8b

regimes581.0ms

Accuracy

37.9% (5.4b remaining)

Error of 12.1b against oracle of 6.7b and baseline of 15.3b

bsearch1.2s

end0.0ms

sample13.9s

Algorithm
intervals