Average Error: 32.1 → 12.0
Time: 1.5m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.628511039343269 \cdot 10^{-184}:\\ \;\;\;\;\frac{2}{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(t \cdot \tan k\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \le 1.4324896232433092 \cdot 10^{-232}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}{{\ell}^{2} \cdot \cos k} + \frac{{\left(\sin k\right)}^{2} \cdot {t}^{3}}{{\ell}^{2} \cdot \cos k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)}{\cos k}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -1.628511039343269e-184

    1. Initial program 27.2

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

    3. Applied unpow327.2

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

    4. Applied times-frac19.2

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

    5. Applied associate-*l*17.1

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

    7. Applied pow117.1

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

    8. Applied pow117.1

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

    9. Applied pow117.1

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

    10. Applied pow117.1

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

    11. Applied pow-prod-down17.1

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

    12. Applied pow117.1

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

    13. Applied pow-prod-down17.1

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

    14. Applied pow-prod-down17.1

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

    15. Applied pow-prod-down17.1

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

    16. Simplified11.8

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

    18. Applied *-un-lft-identity11.8

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

    19. Applied associate-/r*11.8

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

    20. Simplified7.4

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

    if -1.628511039343269e-184 < t < 1.4324896232433092e-232

    1. Initial program 62.7

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

    3. Applied unpow362.7

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

    4. Applied times-frac62.6

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

    5. Applied associate-*l*62.6

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

    6. Taylor expanded around -inf 42.5

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

    if 1.4324896232433092e-232 < t

    1. Initial program 29.8

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

    3. Applied unpow329.8

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

    4. Applied times-frac21.7

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

    5. Applied associate-*l*19.8

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

    7. Applied pow119.8

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

    8. Applied pow119.8

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

    9. Applied pow119.8

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

    10. Applied pow119.8

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

    11. Applied pow-prod-down19.8

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

    12. Applied pow119.8

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

    13. Applied pow-prod-down19.8

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

    14. Applied pow-prod-down19.8

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

    15. Applied pow-prod-down19.8

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

    16. Simplified14.4

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

    18. Applied *-un-lft-identity14.4

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

    19. Applied associate-/r*14.4

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

    20. Simplified9.4

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

    22. Applied tan-quot9.5

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

    23. Applied associate-*r/9.5

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

    24. Applied associate-*l/9.5

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

    25. Applied associate-*r/9.5

      \[\leadsto \frac{\frac{2}{1}}{\color{blue}{\frac{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*\right) \cdot \left(\left(t \cdot \sin k\right) \cdot \frac{t}{\ell}\right)}{\cos k}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.628511039343269 \cdot 10^{-184}:\\ \;\;\;\;\frac{2}{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \left(\left(t \cdot \tan k\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \le 1.4324896232433092 \cdot 10^{-232}:\\ \;\;\;\;\frac{2}{\frac{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}{{\ell}^{2} \cdot \cos k} + \frac{{\left(\sin k\right)}^{2} \cdot {t}^{3}}{{\ell}^{2} \cdot \cos k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\sin k \cdot t\right) \cdot \frac{t}{\ell}\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_* \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +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: 1.5m)Debug log

sample399.0ms

Algorithm
intervals

simplify101.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

101.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 29.1b

localize52.0ms

Local error

Found 4 expressions with local error:

13.5b
(/ (pow t 3) (* l l))
12.3b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
3.4b
(* (/ (pow t 3) (* l l)) (sin k))
2.2b
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))

rewrite92.0ms

Algorithm
rewrite-expression-head
Counts
4 → 120
Calls

4 calls. Slowest were:

65.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
13.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
7.0ms
(/ (pow t 3) (* l l))

series574.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

306.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
186.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
65.0ms
(* (/ (pow t 3) (* l l)) (sin k))
18.0ms
(/ (pow t 3) (* l l))

simplify16.0s

Counts
98 → 132
Calls

98 calls. Slowest were:

1.3s
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
762.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
714.0ms
(* (* (* l l) (cos k)) (- (+ 1 (pow (/ k t) 2)) 1))

prune2.8s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 12.2b

localize33.0ms

Local error

Found 4 expressions with local error:

12.3b
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
8.6b
(/ (* t t) l)
4.0b
(* (/ t l) (sin k))
2.2b
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))

rewrite130.0ms

Algorithm
rewrite-expression-head
Counts
4 → 115
Calls

4 calls. Slowest were:

69.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
48.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
7.0ms
(* (/ t l) (sin k))

series661.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

392.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
171.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
81.0ms
(* (/ t l) (sin k))
17.0ms
(/ (* t t) l)

simplify22.1s

Counts
99 → 127
Calls

99 calls. Slowest were:

1.3s
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
725.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
664.0ms
(* (* (* l l) (cos k)) (- (+ 1 (pow (/ k t) 2)) 1))

prune2.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 6.9b

localize34.0ms

Local error

Found 4 expressions with local error:

14.2b
(* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
4.0b
(* (tan k) (/ t l))
0.3b
(* (* (tan k) (/ t l)) (* (/ t l) t))
0.2b
(/ 2 (pow (* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2))) 1))

rewrite93.0ms

Algorithm
rewrite-expression-head
Counts
4 → 105
Calls

4 calls. Slowest were:

65.0ms
(* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
12.0ms
(* (* (tan k) (/ t l)) (* (/ t l) t))
10.0ms
(/ 2 (pow (* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2))) 1))

series1.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

767.0ms
(/ 2 (pow (* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2))) 1))
226.0ms
(* (* (* (tan k) (/ t l)) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
124.0ms
(* (* (tan k) (/ t l)) (* (/ t l) t))
110.0ms
(* (tan k) (/ t l))

simplify11.7s

Counts
75 → 117
Calls

75 calls. Slowest were:

643.0ms
(* (* (* (sin k) t) (* (/ t l) t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
617.0ms
(* (* (* (tan k) (/ t l)) (* t t)) (* (sin k) (fma (/ k t) (/ k t) 2)))
466.0ms
(* (* (* (sin k) (/ t l)) (* t t)) (* (sin k) (fma (/ k t) (/ k t) 2)))

prune1.9s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 6.3b

localize18.0ms

Local error

Found 4 expressions with local error:

9.5b
(* (* (* (/ t l) (sin k)) (fma (/ k t) (/ k t) 2)) (* (* t (tan k)) (/ t l)))
5.1b
(* (* (/ t l) (sin k)) (fma (/ k t) (/ k t) 2))
4.0b
(* (/ t l) (sin k))
1.3b
(* (* t (tan k)) (/ t l))

rewrite81.0ms

Algorithm
rewrite-expression-head
Counts
4 → 102
Calls

4 calls. Slowest were:

49.0ms
(* (* (* (/ t l) (sin k)) (fma (/ k t) (/ k t) 2)) (* (* t (tan k)) (/ t l)))
22.0ms
(* (* (/ t l) (sin k)) (fma (/ k t) (/ k t) 2))
5.0ms
(* (* t (tan k)) (/ t l))

series789.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

373.0ms
(* (* (* (/ t l) (sin k)) (fma (/ k t) (/ k t) 2)) (* (* t (tan k)) (/ t l)))
190.0ms
(* (* (/ t l) (sin k)) (fma (/ k t) (/ k t) 2))
135.0ms
(* (* t (tan k)) (/ t l))
91.0ms
(* (/ t l) (sin k))

simplify9.9s

Counts
73 → 114
Calls

73 calls. Slowest were:

787.0ms
(* (* (* (* (* (/ t l) (/ t l)) (/ t l)) (* (* (sin k) (sin k)) (sin k))) (* (* (fma (/ k t) (/ k t) 2) (fma (/ k t) (/ k t) 2)) (fma (/ k t) (/ k t) 2))) (* (* (* (* t (tan k)) (* t (tan k))) (* t (tan k))) (* (* (/ t l) (/ t l)) (/ t l))))
579.0ms
(* (* (* t (sin k)) (fma (/ k t) (/ k t) 2)) (* (* t (sin k)) (/ t l)))
492.0ms
(* (* (* (* t (tan k)) (* t (tan k))) (* t (tan k))) (* (* (/ t l) (/ t l)) (/ t l)))

prune1.8s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 6.3b

regimes457.0ms

Accuracy

16% (4.4b remaining)

Error of 12.0b against oracle of 7.6b and baseline of 12.8b

bsearch1.4s

end0.0ms

sample16.1s

Algorithm
intervals