Average Error: 46.9 → 1.3
Time: 8.4m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{\ell}{k} \cdot \sqrt[3]{2}}{t \cdot \sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\tan k}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 46.9

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

  2. Initial simplification30.4

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
  3. Using strategy rm

  4. Applied times-frac29.4

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

  5. Applied add-cube-cbrt29.5

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

  6. Applied times-frac29.1

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

  7. Applied times-frac18.1

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]

  8. Simplified10.8

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
  9. Using strategy rm

  10. Applied div-inv10.8

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\color{blue}{k \cdot \frac{1}{t}}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  11. Applied div-inv10.8

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\color{blue}{\ell \cdot \frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  12. Applied *-un-lft-identity10.8

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\color{blue}{1 \cdot \sin k}}{\ell \cdot \frac{1}{t}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  13. Applied times-frac11.3

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\color{blue}{\frac{1}{\ell} \cdot \frac{\sin k}{\frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  14. Applied times-frac11.7

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  15. Applied times-frac7.8

    \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{\ell}}}{k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}}\right)} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  16. Applied associate-*l*7.5

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{\ell}}}{k} \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\right)}\]

  17. Simplified7.5

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}} \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\right)\]
  18. Using strategy rm

  19. Applied cbrt-div7.5

    \[\leadsto \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\frac{\sqrt[3]{2}}{\sqrt[3]{\tan k}}}\right)\right)\]

  20. Applied associate-*r/2.6

    \[\leadsto \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}} \cdot \left(\color{blue}{\frac{\frac{1}{k} \cdot \ell}{t}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{\tan k}}\right)\right)\]

  21. Applied frac-times2.3

    \[\leadsto \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}} \cdot \color{blue}{\frac{\left(\frac{1}{k} \cdot \ell\right) \cdot \sqrt[3]{2}}{t \cdot \sqrt[3]{\tan k}}}\right)\]

  22. Applied frac-times1.8

    \[\leadsto \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}} \cdot \left(\left(\frac{1}{k} \cdot \ell\right) \cdot \sqrt[3]{2}\right)}{\frac{1}{t} \cdot \left(t \cdot \sqrt[3]{\tan k}\right)}}\]

  23. Simplified1.3

    \[\leadsto \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\color{blue}{\frac{\sqrt[3]{2} \cdot \frac{\ell}{k}}{t \cdot \sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}}}{\frac{1}{t} \cdot \left(t \cdot \sqrt[3]{\tan k}\right)}\]

  24. Simplified1.3

    \[\leadsto \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\frac{\sqrt[3]{2} \cdot \frac{\ell}{k}}{t \cdot \sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\color{blue}{\sqrt[3]{\tan k}}}\]

  25. Final simplification1.3

    \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \sqrt[3]{2}}{t \cdot \sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\tan k}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}\]

Reproduce

herbie shell --seed 2018362 
(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: 7.7m)Debug log

start1.7s

Algorithm
intervals

setup526.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 31.7b

localize68.0ms

Local error

Found 4 expressions with local error:

31.3b
(/ (/ (/ 2 (tan k)) (/ (* (sin k) t) (* (/ l t) (/ l t)))) (* (/ k t) (/ k t)))
1.3b
(/ (/ 2 (tan k)) (/ (* (sin k) t) (* (/ l t) (/ l t))))
0.7b
(/ (* (sin k) t) (* (/ l t) (/ l t)))
0.2b
(* (sin k) t)

rewrite29.0ms

Algorithm
rewrite-expression-head
Counts
4 → 223
Calls

4 calls. Slowest were:

11.0ms
(/ (/ (/ 2 (tan k)) (/ (* (sin k) t) (* (/ l t) (/ l t)))) (* (/ k t) (/ k t)))
7.0ms
(/ (/ 2 (tan k)) (/ (* (sin k) t) (* (/ l t) (/ l t))))
5.0ms
(/ (* (sin k) t) (* (/ l t) (/ l t)))

series718.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

298.0ms
(/ (/ (/ 2 (tan k)) (/ (* (sin k) t) (* (/ l t) (/ l t)))) (* (/ k t) (/ k t)))
251.0ms
(/ (/ 2 (tan k)) (/ (* (sin k) t) (* (/ l t) (/ l t))))
122.0ms
(/ (* (sin k) t) (* (/ l t) (/ l t)))
47.0ms
(* (sin k) t)

simplify58.5s

Counts
272 → 235
Calls

272 calls. Slowest were:

873.0ms
(/ (/ 2 (tan k)) (/ 1 (* (/ l t) (/ l t))))
764.0ms
(/ (/ (/ 1 (tan k)) (* t t)) (/ k t))
750.0ms
(/ (/ (/ 2 (tan k)) t) (/ k t))

prune3.6s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 4.1b

localize24.0ms

Local error

Found 4 expressions with local error:

14.3b
(/ (/ (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k)))) (/ (sin k) (/ l t))) (/ k t))
5.9b
(* (/ 1 k) (/ l t))
3.2b
(/ (sin k) (/ l t))
1.3b
(* (/ (/ (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k)))) (/ (sin k) (/ l t))) (/ k t)) (* (* (/ 1 k) (/ l t)) (cbrt (/ 2 (tan k)))))

rewrite71.0ms

Algorithm
rewrite-expression-head
Counts
4 → 346
Calls

4 calls. Slowest were:

38.0ms
(* (/ (/ (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k)))) (/ (sin k) (/ l t))) (/ k t)) (* (* (/ 1 k) (/ l t)) (cbrt (/ 2 (tan k)))))
12.0ms
(/ (/ (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k)))) (/ (sin k) (/ l t))) (/ k t))
3.0ms
(* (/ 1 k) (/ l t))

series710.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

375.0ms
(/ (/ (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k)))) (/ (sin k) (/ l t))) (/ k t))
252.0ms
(* (/ (/ (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k)))) (/ (sin k) (/ l t))) (/ k t)) (* (* (/ 1 k) (/ l t)) (cbrt (/ 2 (tan k)))))
70.0ms
(/ (sin k) (/ l t))
12.0ms
(* (/ 1 k) (/ l t))

simplify1.0m

Counts
420 → 358
Calls

420 calls. Slowest were:

1.0s
(* (/ (/ (* (* (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k)))) (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k))))) (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k))))) (/ (* (* (sin k) (sin k)) (sin k)) (* (* (/ l t) (/ l t)) (/ l t)))) (* (* (/ k t) (/ k t)) (/ k t))) (* (* (* (* (/ 1 k) (/ l t)) (* (/ 1 k) (/ l t))) (* (/ 1 k) (/ l t))) (/ 2 (tan k))))
933.0ms
(/ (/ (cbrt (/ 2 (tan k))) (/ (sqrt (sin k)) (/ l t))) (/ 1 t))
735.0ms
(* (/ (/ (* (* (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k)))) (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k))))) (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k))))) (* (* (/ (sin k) (/ l t)) (/ (sin k) (/ l t))) (/ (sin k) (/ l t)))) (* (* (/ k t) (/ k t)) (/ k t))) (* (* (* (* (/ 1 k) (/ l t)) (* (/ 1 k) (/ l t))) (* (/ 1 k) (/ l t))) (/ 2 (tan k))))

prune7.4s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 3.4b

localize56.0ms

Local error

Found 4 expressions with local error:

10.2b
(cbrt (/ (sin k) (/ l t)))
10.2b
(cbrt (/ (sin k) (/ l t)))
10.2b
(cbrt (/ (sin k) (/ l t)))
5.9b
(* (/ 1 k) (/ l t))

rewrite14.0ms

Algorithm
rewrite-expression-head
Counts
4 → 100
Calls

4 calls. Slowest were:

6.0ms
(* (/ 1 k) (/ l t))
2.0ms
(cbrt (/ (sin k) (/ l t)))
2.0ms
(cbrt (/ (sin k) (/ l t)))

series374.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

135.0ms
(cbrt (/ (sin k) (/ l t)))
123.0ms
(cbrt (/ (sin k) (/ l t)))
100.0ms
(cbrt (/ (sin k) (/ l t)))
15.0ms
(* (/ 1 k) (/ l t))

simplify2.8s

Counts
78 → 112
Calls

78 calls. Slowest were:

400.0ms
(* (* (* (/ 1 k) (/ 1 k)) (/ 1 k)) (* (* (/ l t) (/ l t)) (/ l t)))
289.0ms
(exp (* 1/3 (- (+ (log (sin k)) (log (/ 1 l))) (log (/ 1 t)))))
269.0ms
(exp (* 1/3 (- (+ (log (sin k)) (log (/ 1 l))) (log (/ 1 t)))))

prune2.7s

Pruning

16 alts after pruning (15 fresh and 1 done)

Merged error: 3.4b

localize43.0ms

Local error

Found 4 expressions with local error:

5.9b
(* (/ 1 k) (/ l t))
1.2b
(* (/ (/ (cbrt (/ 2 (tan k))) (/ (sin k) (/ 1 t))) (/ 1 t)) (* (* (/ 1 k) (/ l t)) (cbrt (/ 2 (tan k)))))
1.0b
(/ (/ (cbrt (/ 2 (tan k))) (/ (sin k) (/ 1 t))) (/ 1 t))
0.5b
(cbrt (/ 2 (tan k)))

rewrite144.0ms

Algorithm
rewrite-expression-head
Counts
4 → 1383
Calls

4 calls. Slowest were:

46.0ms
(* (/ (/ (cbrt (/ 2 (tan k))) (/ (sin k) (/ 1 t))) (/ 1 t)) (* (* (/ 1 k) (/ l t)) (cbrt (/ 2 (tan k)))))
13.0ms
(/ (/ (cbrt (/ 2 (tan k))) (/ (sin k) (/ 1 t))) (/ 1 t))
6.0ms
(* (/ 1 k) (/ l t))

series947.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

478.0ms
(* (/ (/ (cbrt (/ 2 (tan k))) (/ (sin k) (/ 1 t))) (/ 1 t)) (* (* (/ 1 k) (/ l t)) (cbrt (/ 2 (tan k)))))
237.0ms
(cbrt (/ 2 (tan k)))
213.0ms
(/ (/ (cbrt (/ 2 (tan k))) (/ (sin k) (/ 1 t))) (/ 1 t))
19.0ms
(* (/ 1 k) (/ l t))

simplify5.0m

Counts
1925 → 1395
Calls

1925 calls. Slowest were:

6.5s
(/ (/ (cbrt (sqrt (/ 2 (tan k)))) (/ 1 (* (cbrt (/ 1 t)) (cbrt (/ 1 t))))) (* (cbrt (/ 1 t)) (cbrt (/ 1 t))))
6.4s
(/ (/ (sqrt (cbrt (/ 2 (tan k)))) (/ 1 (* (cbrt (/ 1 t)) (cbrt (/ 1 t))))) (* (cbrt (/ 1 t)) (cbrt (/ 1 t))))
6.3s
(/ (/ (cbrt (* (cbrt (/ 2 (tan k))) (cbrt (/ 2 (tan k))))) (/ 1 (* (cbrt (/ 1 t)) (cbrt (/ 1 t))))) (* (cbrt (/ 1 t)) (cbrt (/ 1 t))))

prune19.6s

Pruning

18 alts after pruning (17 fresh and 1 done)

Merged error: 0.1b

regimes450.0ms

Accuracy

0% (1.2b remaining)

Error of 1.3b against oracle of 0.1b and baseline of 1.3b

bsearch3.0ms