Average Error: 31.8 → 8.1
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.089553883996185 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \sin k\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 15389.415660968743:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{t \cdot t}{\frac{\ell}{2}} + \frac{k}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \sin k\right)}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right) \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \end{array}\]

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 3 regimes

  2. if t < -4.089553883996185e-49

    1. Initial program 21.6

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

    3. Applied unpow321.6

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

    4. Applied times-frac16.0

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

    5. Applied associate-*l*13.4

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

    7. Applied associate-/l*7.7

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

    9. Applied tan-quot7.7

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

    10. Applied associate-*l/6.5

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

    11. Applied frac-times4.7

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

    12. Applied associate-*l/4.2

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

    14. Applied associate-*l/4.2

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

    if -4.089553883996185e-49 < t < 15389.415660968743

    1. Initial program 51.3

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

    3. Applied unpow351.3

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

    4. Applied times-frac42.9

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

    5. Applied associate-*l*41.7

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

    7. Applied associate-/l*35.8

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

    9. Applied tan-quot35.8

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

    10. Applied associate-*l/35.8

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

    11. Applied frac-times37.3

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

    12. Applied associate-*l/34.1

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

    13. Taylor expanded around inf 21.7

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

    14. Simplified16.3

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

    if 15389.415660968743 < t

    1. Initial program 21.1

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

    3. Applied unpow321.1

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

    4. Applied times-frac15.0

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

    5. Applied associate-*l*13.3

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

    7. Applied associate-/l*7.5

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

    9. Applied associate-*l/6.1

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

    10. Applied associate-*l/3.4

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

    11. Applied associate-*l/3.1

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.089553883996185 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \frac{\sin k \cdot t}{\ell}\right) \cdot \sin k\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;t \le 15389.415660968743:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{t \cdot t}{\frac{\ell}{2}} + \frac{k}{\frac{\ell}{k}}\right) \cdot \left(\sin k \cdot \sin k\right)}{\cos k \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t\right) \cdot \tan k\right)}{\frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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: 1.8m)Debug log

sample508.0ms

Algorithm
intervals

simplify164.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

164.0ms
(/ 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: 34.8b

localize73.0ms

Local error

Found 4 expressions with local error:

16.3b
(/ (pow t 3) (* l l))
14.3b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
5.0b
(* (/ (pow t 3) (* l l)) (sin k))
3.4b
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))

rewrite127.0ms

Algorithm
rewrite-expression-head
Counts
4 → 112
Calls

4 calls. Slowest were:

82.0ms
(* (* (* (/ (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))

series654.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

343.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
195.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
87.0ms
(* (/ (pow t 3) (* l l)) (sin k))
28.0ms
(/ (pow t 3) (* l l))

simplify16.4s

Counts
98 → 124
Calls

98 calls. Slowest were:

786.0ms
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
746.0ms
(* (cos k) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
709.0ms
(* (* l l) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))

prune2.1s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 18.5b

localize24.0ms

Local error

Found 4 expressions with local error:

14.3b
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
8.2b
(/ (* t t) l)
3.4b
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
2.9b
(* (/ t l) (sin k))

rewrite168.0ms

Algorithm
rewrite-expression-head
Counts
4 → 107
Calls

4 calls. Slowest were:

116.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
42.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
4.0ms
(/ (* t t) l)

series601.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

347.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
168.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
76.0ms
(* (/ t l) (sin k))
10.0ms
(/ (* t t) l)

simplify22.1s

Counts
99 → 119
Calls

99 calls. Slowest were:

1.1s
(* (* l (cos k)) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
790.0ms
(* (* l l) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
749.0ms
(* (* l (cos k)) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))

prune2.0s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 12.4b

localize37.0ms

Local error

Found 4 expressions with local error:

14.3b
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
3.4b
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
2.9b
(* (/ t l) (sin k))
2.4b
(* (/ t (/ l t)) (* (/ t l) (sin k)))

rewrite146.0ms

Algorithm
rewrite-expression-head
Counts
4 → 119
Calls

4 calls. Slowest were:

96.0ms
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
23.0ms
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
19.0ms
(* (/ t (/ l t)) (* (/ t l) (sin k)))

series744.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

391.0ms
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
187.0ms
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
88.0ms
(* (/ t (/ l t)) (* (/ t l) (sin k)))
78.0ms
(* (/ t l) (sin k))

simplify24.4s

Counts
112 → 131
Calls

112 calls. Slowest were:

829.0ms
(* (* l (cos k)) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
820.0ms
(* (cos k) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
805.0ms
(* (* (* t (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))

prune2.2s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 8.4b

localize30.0ms

Local error

Found 4 expressions with local error:

10.1b
(* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1))
7.5b
(/ (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (/ l t) (cos k)))
2.9b
(* (/ t l) (sin k))
2.8b
(* (* t (* (/ t l) (sin k))) (sin k))

rewrite86.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

38.0ms
(/ (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (/ l t) (cos k)))
31.0ms
(* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1))
10.0ms
(* (* t (* (/ t l) (sin k))) (sin k))

series658.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

279.0ms
(/ (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (/ l t) (cos k)))
229.0ms
(* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1))
75.0ms
(* (/ t l) (sin k))
74.0ms
(* (* t (* (/ t l) (sin k))) (sin k))

simplify15.8s

Counts
68 → 103
Calls

68 calls. Slowest were:

1.0s
(/ (* (* (* (* (* t (* (/ t l) (sin k))) (* t (* (/ t l) (sin k)))) (* t (* (/ t l) (sin k)))) (* (* (sin k) (sin k)) (sin k))) (* (* (+ (+ 1 (pow (/ k t) 2)) 1) (+ (+ 1 (pow (/ k t) 2)) 1)) (+ (+ 1 (pow (/ k t) 2)) 1))) (* (* (* (/ l t) (/ l t)) (/ l t)) (* (* (cos k) (cos k)) (cos k))))
1.0s
(/ (* (* (* (* (* t (* (/ t l) (sin k))) (* t (* (/ t l) (sin k)))) (* t (* (/ t l) (sin k)))) (* (* (sin k) (sin k)) (sin k))) (* (* (+ (+ 1 (pow (/ k t) 2)) 1) (+ (+ 1 (pow (/ k t) 2)) 1)) (+ (+ 1 (pow (/ k t) 2)) 1))) (* (* (* (/ l t) (cos k)) (* (/ l t) (cos k))) (* (/ l t) (cos k))))
978.0ms
(/ (* (* (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1))) (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1))) (* (* (* (/ l t) (/ l t)) (/ l t)) (* (* (cos k) (cos k)) (cos k))))

prune1.8s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 6.1b

regimes321.0ms

Accuracy

66.7% (3.1b remaining)

Error of 8.1b against oracle of 5.0b and baseline of 14.2b

bsearch991.0ms

end0.0ms

sample15.7s

Algorithm
intervals