Average Error: 47.4 → 1.8
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{2}{\sin k \cdot \left(\frac{\sin k}{\frac{\ell}{k}} \cdot \frac{\frac{t}{\cos k}}{\frac{\ell}{k}}\right)}\]

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 47.4

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

  2. Taylor expanded around inf 23.9

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

  4. Applied associate-/l*24.0

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

  6. Applied times-frac22.7

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

  7. Applied associate-/r*21.9

    \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t}{\frac{{\ell}^{2}}{{k}^{2}}}}{\frac{\cos k}{{\left(\sin k\right)}^{2}}}}}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt42.1

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

  10. Applied unpow-prod-down42.1

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

  11. Applied *-un-lft-identity42.1

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

  12. Applied times-frac42.1

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

  13. Applied *-un-lft-identity42.1

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

  14. Applied times-frac42.1

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

  15. Simplified42.1

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

  16. Simplified1.8

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

  17. Final simplification1.8

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

Reproduce

herbie shell --seed 2019022 
(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: 2.1m)Debug log

sample1.6s

Algorithm
intervals
Results
779.0ms136×body1280valid
334.0ms77×body2560valid
276.0ms34×body5120valid
115.0ms70×body640valid
48.0ms40×body320valid
23.0ms62×body80valid
13.0ms17×body160valid

simplify132.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
132.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1)))

prune9.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 47.1b

localize38.0ms

Local error

Found 4 expressions with local error:

32.7b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
16.1b
(/ (pow t 3) (* l l))
1.4b
(- (+ 1 (pow (/ k t) 2)) 1)
1.1b
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))

rewrite65.0ms

Algorithm
rewrite-expression-head
Rules
37×add-exp-log
29×add-cbrt-cube
19×prod-exp
18×pow1
15×associate-*l/
14×cbrt-unprod
10×frac-times
10×add-sqr-sqrt
9×pow-prod-down
8×add-cube-cbrt
8×tan-quot
8×*-un-lft-identity
7×associate-*r*
6×associate-*r/
6×div-exp
5×associate-/l*
5×flip--
5×flip3--
5×times-frac
4×add-log-exp
3×associate--l+
3×distribute-lft-in
3×sub-neg
3×cbrt-undiv
3×pow-to-exp
3×distribute-rgt-in
2×*-commutative
2×associate-*l*
2×difference-of-sqr-1
2×unpow3
2×cube-mult
1×div-inv
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 103
Calls
4 calls:
Slowest
43.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
11.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
7.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
3.0ms
(/ (pow t 3) (* l l))

series607.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
387.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
172.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
25.0ms
(/ (pow t 3) (* l l))
22.0ms
(- (+ 1 (pow (/ k t) 2)) 1)

simplify17.3s

Counts
86 → 115
Calls
86 calls:
Slowest
862.0ms
(* (* (* (pow t 3) (sin k)) (sin k)) (- (+ 1 (pow (/ k t) 2)) 1))
848.0ms
(* (* (* (pow t 3) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
645.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
629.0ms
(sqrt (- (+ 1 (pow (/ k t) 2)) 1))
581.0ms
(* (* (* (/ (* (* (pow t 3) (pow t 3)) (pow t 3)) (* (* (* l l) (* l l)) (* l l))) (* (* (sin k) (sin k)) (sin k))) (* (* (tan k) (tan k)) (tan k))) (* (* (- (+ 1 (pow (/ k t) 2)) 1) (- (+ 1 (pow (/ k t) 2)) 1)) (- (+ 1 (pow (/ k t) 2)) 1)))

prune2.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 15.9b

localize40.0ms

Local error

Found 4 expressions with local error:

18.0b
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
5.2b
(* t (* (pow k 2) (pow (sin k) 2)))
0.4b
(pow (sin k) 2)
0.3b
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
12×add-exp-log
9×add-cbrt-cube
6×add-cube-cbrt
6×*-un-lft-identity
6×associate-/r*
6×add-sqr-sqrt
5×pow1
4×add-log-exp
3×div-inv
3×div-exp
3×unpow-prod-down
2×prod-exp
2×times-frac
2×frac-2neg
2×clear-num
2×cbrt-undiv
2×pow-to-exp
1×unpow2
1×associate-/l*
1×cbrt-unprod
1×*-commutative
1×associate-/r/
1×pow-exp
1×associate-*r*
1×pow-pow
Counts
4 → 57
Calls
4 calls:
Slowest
9.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
5.0ms
(* t (* (pow k 2) (pow (sin k) 2)))
1.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
1.0ms
(pow (sin k) 2)

series570.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
274.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
211.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
69.0ms
(* t (* (pow k 2) (pow (sin k) 2)))
16.0ms
(pow (sin k) 2)

simplify6.0s

Counts
32 → 69
Calls
32 calls:
Slowest
627.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
552.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
552.0ms
(cbrt (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
547.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
519.0ms
(sqrt (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))

prune1.0s

Pruning

21 alts after pruning (21 fresh and 0 done)

Merged error: 13.1b

localize8.0ms

Local error

Found 4 expressions with local error:

16.2b
(/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2)))
5.1b
(/ t (/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2))))
0.4b
(pow (sin k) 2)
0.3b
(/ 2 (/ t (/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2)))))

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
57×add-exp-log
32×prod-exp
23×pow-to-exp
21×div-exp
16×add-cbrt-cube
11×associate-/r*
7×add-cube-cbrt
7×pow-exp
7×*-un-lft-identity
7×add-sqr-sqrt
5×div-inv
5×pow1
4×add-log-exp
4×cbrt-unprod
4×cbrt-undiv
3×associate-/r/
3×frac-2neg
3×unpow-prod-down
3×clear-num
2×times-frac
1×unpow2
1×associate-/l*
1×pow-pow
Counts
4 → 84
Calls
4 calls:
Slowest
18.0ms
(/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2)))
3.0ms
(/ t (/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2))))
2.0ms
(pow (sin k) 2)
1.0ms
(/ 2 (/ t (/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2)))))

series688.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
302.0ms
(/ t (/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2))))
248.0ms
(/ 2 (/ t (/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2)))))
113.0ms
(/ (* (pow l 2) (cos k)) (* (pow k 2) (pow (sin k) 2)))
24.0ms
(pow (sin k) 2)

simplify12.5s

Counts
58 → 96
Calls
58 calls:
Slowest
515.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
487.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
447.0ms
(- (log (* (pow l 2) (cos k))) (+ (log (pow k 2)) (log (pow (sin k) 2))))
424.0ms
(- (log (* (pow l 2) (cos k))) (+ (* (log k) 2) (* (log (sin k)) 2)))
417.0ms
(- (log (* (pow l 2) (cos k))) (+ (* (log k) 2) (* (log (sin k)) 2)))

prune1.6s

Pruning

22 alts after pruning (22 fresh and 0 done)

Merged error: 10.2b

localize27.0ms

Local error

Found 4 expressions with local error:

14.0b
(/ (pow l 2) (pow k 2))
3.9b
(/ (/ t (/ (pow l 2) (pow k 2))) (/ (cos k) (pow (sin k) 2)))
3.9b
(/ t (/ (pow l 2) (pow k 2)))
0.4b
(pow (sin k) 2)

rewrite29.0ms

Algorithm
rewrite-expression-head
Rules
283×times-frac
134×add-cube-cbrt
134×*-un-lft-identity
134×add-sqr-sqrt
57×unpow-prod-down
49×associate-/r*
37×unpow2
36×div-inv
28×associate-/r/
19×add-exp-log
11×div-exp
11×add-cbrt-cube
9×associate-/l*
6×pow-to-exp
5×pow1
4×add-log-exp
4×cbrt-undiv
3×frac-2neg
3×clear-num
2×pow-exp
1×associate-/l/
1×pow-pow
Counts
4 → 257
Calls
4 calls:
Slowest
10.0ms
(/ (/ t (/ (pow l 2) (pow k 2))) (/ (cos k) (pow (sin k) 2)))
6.0ms
(/ (pow l 2) (pow k 2))
2.0ms
(pow (sin k) 2)
2.0ms
(/ t (/ (pow l 2) (pow k 2)))

series789.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
697.0ms
(/ (/ t (/ (pow l 2) (pow k 2))) (/ (cos k) (pow (sin k) 2)))
55.0ms
(/ t (/ (pow l 2) (pow k 2)))
22.0ms
(/ (pow l 2) (pow k 2))
15.0ms
(pow (sin k) 2)

simplify33.5s

Counts
365 → 269
Calls
365 calls:
Slowest
845.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
566.0ms
(/ (/ t (pow l 2)) (/ 1 (pow (sqrt (sin k)) 2)))
511.0ms
(/ (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (pow k 2) (pow k 2)) (pow k 2)))
509.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
487.0ms
(/ (* (cbrt (/ t (/ (pow l 2) (pow k 2)))) (cbrt (/ t (/ (pow l 2) (pow k 2))))) (/ (* (cbrt (cos k)) (cbrt (cos k))) (* (cbrt (pow (sin k) 2)) (cbrt (pow (sin k) 2)))))

prune4.0s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 0.1b

regimes450.0ms

Accuracy

0% (1.4b remaining)

Error of 1.8b against oracle of 0.5b and baseline of 1.8b

bsearch1.0ms

end0.0ms

sample40.1s

Algorithm
intervals
Results
10.9s4061×body1280valid
10.2s2198×body2560valid
8.0s959×body5120valid
6.9s2510×body640valid
1.8s1036×body320valid
1.5s2050×body80valid
370.0ms417×body160valid