Average Error: 47.0 → 17.1
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 3.8660636787078 \cdot 10^{-320}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{k \cdot \left(t \cdot k\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\\ \mathbf{elif}\;\ell \cdot \ell \le 1.4084948583551732 \cdot 10^{+131}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}}\\ \mathbf{elif}\;\ell \cdot \ell \le 6.069165899924277 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{{\ell}^{2}}}}{\frac{{\left(\sin k\right)}^{2} \cdot {k}^{2}}{\cos k}}\\ \mathbf{elif}\;\ell \cdot \ell \le 1.8570723857001577 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{2}{{\left(\sin k\right)}^{2} \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\frac{1}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\\ \end{array}\]

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 5 regimes

  2. if (* l l) < 3.8660636787078e-320

    1. Initial program 44.8

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

    2. Taylor expanded around -inf 20.7

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

    4. Applied associate-*r*19.9

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

    6. Applied unpow219.9

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

    7. Applied associate-*r*19.9

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

    9. Applied add-cbrt-cube20.0

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

    10. Applied add-cbrt-cube22.4

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

    11. Applied cbrt-undiv22.4

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

    12. Simplified15.4

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

    if 3.8660636787078e-320 < (* l l) < 1.4084948583551732e+131

    1. Initial program 42.5

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

    2. Taylor expanded around -inf 11.3

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

    4. Applied associate-*r*8.8

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

    6. Applied unpow28.8

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

    7. Applied associate-*r*5.9

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

    9. Applied associate-/l*3.5

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

    if 1.4084948583551732e+131 < (* l l) < 6.069165899924277e+222

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 21.3

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

    4. Applied times-frac21.1

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

    5. Applied associate-/r*21.6

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

    if 6.069165899924277e+222 < (* l l) < 1.8570723857001577e+300

    1. Initial program 49.2

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

    2. Taylor expanded around -inf 30.3

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

    4. Applied associate-*r*30.1

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

    6. Applied unpow230.1

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

    7. Applied associate-*r*21.8

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

    9. Applied div-inv21.8

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

    10. Applied associate-/r*21.8

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

    if 1.8570723857001577e+300 < (* l l)

    1. Initial program 62.0

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

    3. Applied unpow362.0

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

    4. Applied times-frac52.1

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

    5. Applied associate-*l*52.1

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 3.8660636787078 \cdot 10^{-320}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\frac{k \cdot \left(t \cdot k\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right)}^{3}}}\\ \mathbf{elif}\;\ell \cdot \ell \le 1.4084948583551732 \cdot 10^{+131}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}}}\\ \mathbf{elif}\;\ell \cdot \ell \le 6.069165899924277 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{{\ell}^{2}}}}{\frac{{\left(\sin k\right)}^{2} \cdot {k}^{2}}{\cos k}}\\ \mathbf{elif}\;\ell \cdot \ell \le 1.8570723857001577 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{2}{{\left(\sin k\right)}^{2} \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\frac{1}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t \cdot t}{\ell}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\\ \end{array}\]

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: 1.6m)Debug log

sample1.1s

Algorithm
intervals

simplify96.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

96.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1)))

prune22.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 49.0b

localize55.0ms

Local error

Found 4 expressions with local error:

36.6b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
13.3b
(/ (pow t 3) (* l l))
0.8b
(* (/ (pow t 3) (* l l)) (sin k))
0.8b
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))

rewrite66.0ms

Algorithm
rewrite-expression-head
Counts
4 → 125
Calls

4 calls. Slowest were:

43.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
12.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
5.0ms
(* (/ (pow t 3) (* l l)) (sin k))

series566.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

283.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
189.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
74.0ms
(* (/ (pow t 3) (* l l)) (sin k))
20.0ms
(/ (pow t 3) (* l l))

simplify21.8s

Counts
103 → 137
Calls

103 calls. Slowest were:

1.3s
(* (* (* (pow t 3) (sin k)) (sin k)) (- (+ 1 (pow (/ k t) 2)) 1))
1.0s
(sqrt (- (+ 1 (pow (/ k t) 2)) 1))
1.0s
(cbrt (- (+ 1 (pow (/ k t) 2)) 1))

prune2.3s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 22.0b

localize55.0ms

Local error

Found 4 expressions with local error:

22.7b
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
5.8b
(* t (* (pow k 2) (pow (sin k) 2)))
0.4b
(pow (sin k) 2)
0.4b
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))

rewrite34.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

19.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
10.0ms
(* t (* (pow k 2) (pow (sin k) 2)))
3.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))

series591.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

273.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
200.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
93.0ms
(* t (* (pow k 2) (pow (sin k) 2)))
25.0ms
(pow (sin k) 2)

simplify6.1s

Counts
32 → 77
Calls

32 calls. Slowest were:

815.0ms
(/ (* (* (* 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))))
542.0ms
(cbrt (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
520.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))

prune1.3s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 17.6b

localize19.0ms

Local error

Found 4 expressions with local error:

22.7b
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
6.6b
(* t (pow k 2))
0.4b
(pow (sin k) 2)
0.4b
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))

rewrite15.0ms

Algorithm
rewrite-expression-head
Counts
4 → 79
Calls

4 calls. Slowest were:

10.0ms
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
1.0ms
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))
1.0ms
(* t (pow k 2))

series549.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

300.0ms
(/ 2 (/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k))))
200.0ms
(/ (* (* t (pow k 2)) (pow (sin k) 2)) (* (pow l 2) (cos k)))
26.0ms
(pow (sin k) 2)
23.0ms
(* t (pow k 2))

simplify6.0s

Counts
46 → 91
Calls

46 calls. Slowest were:

629.0ms
(/ (* (* (* (* 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) (cos k)) (* (pow l 2) (cos k))) (* (pow l 2) (cos k))))
602.0ms
(/ (* (* (* (* 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))))
584.0ms
(/ (* (* (* (* 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))))

prune1.7s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 15.8b

localize26.0ms

Local error

Found 4 expressions with local error:

22.7b
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
0.4b
(pow (sin k) 2)
0.4b
(/ 2 (/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k))))
0.2b
(* (pow l 2) (cos k))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 85
Calls

4 calls. Slowest were:

24.0ms
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
3.0ms
(* (pow l 2) (cos k))
2.0ms
(/ 2 (/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k))))

series555.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

259.0ms
(/ 2 (/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k))))
221.0ms
(/ (* (* (* t k) k) (pow (sin k) 2)) (* (pow l 2) (cos k)))
48.0ms
(* (pow l 2) (cos k))
26.0ms
(pow (sin k) 2)

simplify6.9s

Counts
51 → 97
Calls

51 calls. Slowest were:

631.0ms
(/ (* (* (* (* (* 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))))
536.0ms
(/ (* (* (* (* (* t k) k) (* (* t k) k)) (* (* t k) k)) (* (* (pow (sin k) 2) (pow (sin k) 2)) (pow (sin k) 2))) (* (* (* (pow l 2) (cos k)) (* (pow l 2) (cos k))) (* (pow l 2) (cos k))))
533.0ms
(/ (* (* (* (* (* t k) k) (pow (sin k) 2)) (* (* (* t k) k) (pow (sin k) 2))) (* (* (* t k) k) (pow (sin k) 2))) (* (* (* (pow l 2) (cos k)) (* (pow l 2) (cos k))) (* (pow l 2) (cos k))))

prune1.6s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 14.9b

regimes696.0ms

Accuracy

33.6% (5.2b remaining)

Error of 17.1b against oracle of 11.9b and baseline of 19.7b

bsearch8.0ms

end0.0ms

sample43.5s

Algorithm
intervals