Average Error: 47.7 → 18.4
Time: 1.5m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -8.927157688158333 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left({k}^{6} \cdot \frac{1}{6} + {k}^{4}\right) \cdot \frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{elif}\;\ell \le -2.7228774398391337 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(\sin k\right)}^{2} \cdot \left(\left(k \cdot t\right) \cdot k\right)} \cdot \left({\ell}^{2} \cdot \cos k\right)\\ \mathbf{elif}\;\ell \le 4.0994613218468663 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{{\ell}^{2}} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \end{array}\]

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. Split input into 4 regimes

  2. if l < -8.927157688158333e+154

    1. Initial program 62.3

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

    2. Taylor expanded around 0 62.4

      \[\leadsto \frac{2}{\color{blue}{\frac{1}{6} \cdot \frac{t \cdot {k}^{6}}{{\ell}^{2}} + \frac{t \cdot {k}^{4}}{{\ell}^{2}}}}\]

    3. Simplified56.2

      \[\leadsto \frac{2}{\color{blue}{\frac{\frac{t}{\ell}}{\ell} \cdot \left({k}^{4} + {k}^{6} \cdot \frac{1}{6}\right)}}\]

    if -8.927157688158333e+154 < l < -2.7228774398391337e-154

    1. Initial program 45.1

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

    2. Taylor expanded around -inf 16.0

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

    4. Applied associate-*r*13.8

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

    6. Applied unpow213.8

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

    7. Applied associate-*r*9.3

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

    9. Applied associate-/r/9.3

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

    if -2.7228774398391337e-154 < l < 4.0994613218468663e-159

    1. Initial program 45.1

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

    2. Taylor expanded around -inf 20.6

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

    4. Applied associate-*r*19.6

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

    6. Applied unpow219.6

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

    7. Applied associate-*r*19.6

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

    9. Applied add-cbrt-cube19.9

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

    10. Applied add-cbrt-cube22.5

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

    11. Applied cbrt-undiv22.5

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

    12. Simplified15.3

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

    if 4.0994613218468663e-159 < l

    1. Initial program 48.9

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

    2. Taylor expanded around -inf 26.0

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

    4. Applied associate-*r*24.7

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

    6. Applied unpow224.7

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

    7. Applied associate-*r*21.6

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

    9. Applied times-frac20.3

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(t \cdot k\right) \cdot k}{{\ell}^{2}} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -8.927157688158333 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left({k}^{6} \cdot \frac{1}{6} + {k}^{4}\right) \cdot \frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{elif}\;\ell \le -2.7228774398391337 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(\sin k\right)}^{2} \cdot \left(\left(k \cdot t\right) \cdot k\right)} \cdot \left({\ell}^{2} \cdot \cos k\right)\\ \mathbf{elif}\;\ell \le 4.0994613218468663 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\left(k \cdot t\right) \cdot k}{\cos k}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot t\right) \cdot k}{{\ell}^{2}} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(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

sample1.6s

Algorithm
intervals

simplify84.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

84.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1)))

prune19.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 46.6b

localize79.0ms

Local error

Found 4 expressions with local error:

34.2b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
12.9b
(/ (pow t 3) (* l l))
3.1b
(- (+ 1 (pow (/ k t) 2)) 1)
0.8b
(* (/ (pow t 3) (* l l)) (sin k))

rewrite116.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

83.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
15.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
11.0ms
(* (/ (pow t 3) (* l l)) (sin k))

series429.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

326.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
73.0ms
(* (/ (pow t 3) (* l l)) (sin k))
17.0ms
(/ (pow t 3) (* l l))
13.0ms
(- (+ 1 (pow (/ k t) 2)) 1)

simplify16.3s

Counts
87 → 118
Calls

87 calls. Slowest were:

843.0ms
(* (* (* (pow t 3) (sin k)) (sin k)) (- (+ 1 (pow (/ k t) 2)) 1))
817.0ms
(* (* (* (pow t 3) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
661.0ms
(sqrt (- (+ 1 (pow (/ k t) 2)) 1))

prune1.9s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 19.0b

localize34.0ms

Local error

Found 4 expressions with local error:

19.4b
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
6.2b
(* t (* (pow k 2) (pow (sin k) 2)))
0.4b
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
0.4b
(pow (sin k) 2)

rewrite17.0ms

Algorithm
rewrite-expression-head
Counts
4 → 57
Calls

4 calls. Slowest were:

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))))

series605.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

256.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
247.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
77.0ms
(* t (* (pow k 2) (pow (sin k) 2)))
25.0ms
(pow (sin k) 2)

simplify5.9s

Counts
32 → 69
Calls

32 calls. Slowest were:

905.0ms
(/ (* (* (* t (* (pow k 2) (pow (sin k) 2))) (* t (* (pow k 2) (pow (sin k) 2)))) (* t (* (pow k 2) (pow (sin k) 2)))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))
543.0ms
(/ (* (* (* t (* (pow k 2) (pow (sin k) 2))) (* t (* (pow k 2) (pow (sin k) 2)))) (* t (* (pow k 2) (pow (sin k) 2)))) (* (* (* (pow l 2) (cos k)) (* (pow l 2) (cos k))) (* (pow l 2) (cos k))))
531.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))

prune1.1s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 15.7b

localize20.0ms

Local error

Found 4 expressions with local error:

19.4b
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
5.8b
(* t (pow k 2))
0.4b
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))
0.4b
(pow (sin k) 2)

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

15.0ms
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
3.0ms
(pow (sin k) 2)
1.0ms
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))

series476.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

273.0ms
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))
171.0ms
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
16.0ms
(* t (pow k 2))
16.0ms
(pow (sin k) 2)

simplify6.5s

Counts
46 → 83
Calls

46 calls. Slowest were:

635.0ms
(- (+ (log (* t (pow k 2))) (* (log (sin k)) 2)) (log (* (pow l 2) (cos k))))
576.0ms
(/ (* (* (* (* t (pow k 2)) (pow (sin k) 2)) (* (* t (pow k 2)) (pow (sin k) 2))) (* (* t (pow k 2)) (pow (sin k) 2))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))
561.0ms
(/ (* (* (* (* t (pow k 2)) (* t (pow k 2))) (* t (pow k 2))) (* (* (pow (sin k) 2) (pow (sin k) 2)) (pow (sin k) 2))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))

prune1.4s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 12.1b

localize13.0ms

Local error

Found 4 expressions with local error:

19.4b
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
0.4b
(/ 2 (/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k))))
0.4b
(pow (sin k) 2)
0.2b
(* (pow l 2) (cos k))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 77
Calls

4 calls. Slowest were:

11.0ms
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
7.0ms
(* (pow l 2) (cos k))
2.0ms
(pow (sin k) 2)

series581.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

289.0ms
(/ 2 (/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k))))
228.0ms
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
49.0ms
(* (pow l 2) (cos k))
15.0ms
(pow (sin k) 2)

simplify6.8s

Counts
51 → 89
Calls

51 calls. Slowest were:

627.0ms
(/ (* (* (* (* (* t k) k) (pow (sin k) 2)) (* (* (* t k) k) (pow (sin k) 2))) (* (* (* t k) k) (pow (sin k) 2))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))
564.0ms
(/ (* (* (* (* (* t k) k) (* (* t k) k)) (* (* t k) k)) (* (* (pow (sin k) 2) (pow (sin k) 2)) (pow (sin k) 2))) (* (* (* (pow l 2) (pow l 2)) (pow l 2)) (* (* (cos k) (cos k)) (cos k))))
519.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))

prune1.1s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 11.4b

regimes470.0ms

Accuracy

19.3% (6.0b remaining)

Error of 18.4b against oracle of 12.4b and baseline of 19.9b

bsearch2.5s

end0.0ms

sample42.5s

Algorithm
intervals