Average Error: 32.1 → 11.4
Time: 1.7m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.678878777389776 \cdot 10^{-24} \lor \neg \left(t \le 1.0092838300072265 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{2}{\sqrt[3]{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\frac{\ell}{t}}\right) \cdot \tan k\right)} \cdot \left(\sqrt[3]{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\frac{\ell}{t}}\right) \cdot \tan k\right)} \cdot \sqrt[3]{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\frac{\ell}{t}}\right) \cdot \tan k\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \sin k\right) \cdot \left(\frac{k}{\frac{\ell}{k}} + \frac{t \cdot t}{\frac{\ell}{2}}\right)}{\cos k \cdot \frac{\ell}{t}}}\\ \end{array}\]

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 2 regimes

  2. if t < -2.678878777389776e-24 or 1.0092838300072265e-59 < t

    1. Initial program 22.6

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

    3. Applied unpow322.6

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

    4. Applied times-frac16.2

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

    5. Applied associate-*l*14.0

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

    7. Applied associate-/l*8.7

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

    9. Applied add-cube-cbrt8.9

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

    if -2.678878777389776e-24 < t < 1.0092838300072265e-59

    1. Initial program 52.9

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

    3. Applied unpow352.9

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

    4. Applied times-frac43.7

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

    5. Applied associate-*l*42.9

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

    7. Applied associate-/l*36.7

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

    9. Applied tan-quot36.7

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

    10. Applied associate-*l/36.7

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

    11. Applied frac-times38.2

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

    12. Applied associate-*l/35.0

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

    13. Taylor expanded around -inf 21.8

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

    14. Simplified16.8

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.678878777389776 \cdot 10^{-24} \lor \neg \left(t \le 1.0092838300072265 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{2}{\sqrt[3]{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\frac{\ell}{t}}\right) \cdot \tan k\right)} \cdot \left(\sqrt[3]{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\frac{\ell}{t}}\right) \cdot \tan k\right)} \cdot \sqrt[3]{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\frac{\ell}{t}}\right) \cdot \tan k\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \sin k\right) \cdot \left(\frac{k}{\frac{\ell}{k}} + \frac{t \cdot t}{\frac{\ell}{2}}\right)}{\cos k \cdot \frac{\ell}{t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(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: 1.7m)Debug log

sample413.0ms

Algorithm
intervals

simplify102.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

101.0ms
(/ 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: 29.1b

localize465.0ms

Local error

Found 4 expressions with local error:

13.5b
(/ (pow t 3) (* l l))
12.3b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
3.4b
(* (/ (pow t 3) (* l l)) (sin k))
2.2b
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))

rewrite77.0ms

Algorithm
rewrite-expression-head
Counts
4 → 112
Calls

4 calls. Slowest were:

44.0ms
(* (* (* (/ (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))
9.0ms
(* (/ (pow t 3) (* l l)) (sin k))

series600.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

342.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
159.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
77.0ms
(* (/ (pow t 3) (* l l)) (sin k))
22.0ms
(/ (pow t 3) (* l l))

simplify14.7s

Counts
98 → 124
Calls

98 calls. Slowest were:

718.0ms
(* (* (* (pow t 3) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
678.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
612.0ms
(* (cos k) (- (+ 1 (pow (/ k t) 2)) 1))

prune2.2s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 13.0b

localize21.0ms

Local error

Found 4 expressions with local error:

12.3b
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
8.6b
(/ (* t t) l)
4.0b
(* (/ t l) (sin k))
2.2b
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))

rewrite148.0ms

Algorithm
rewrite-expression-head
Counts
4 → 107
Calls

4 calls. Slowest were:

87.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
46.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
7.0ms
(* (/ t l) (sin k))

series708.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

419.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
182.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
90.0ms
(* (/ t l) (sin k))
16.0ms
(/ (* t t) l)

simplify20.6s

Counts
99 → 119
Calls

99 calls. Slowest were:

694.0ms
(* (* (* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (* (/ (* t t) l) (* (/ t l) (sin k)))) (* (/ (* t t) l) (* (/ t l) (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)))
676.0ms
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
664.0ms
(* (* l l) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))

prune1.8s

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 8.2b

localize16.0ms

Local error

Found 4 expressions with local error:

12.3b
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
4.0b
(* (/ t l) (sin k))
2.2b
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
0.3b
(* (/ t (/ l t)) (* (/ t l) (sin k)))

rewrite160.0ms

Algorithm
rewrite-expression-head
Counts
4 → 119
Calls

4 calls. Slowest were:

91.0ms
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
40.0ms
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
18.0ms
(* (/ t (/ l t)) (* (/ t l) (sin k)))

series718.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

343.0ms
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
212.0ms
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
86.0ms
(* (/ t (/ l t)) (* (/ t l) (sin k)))
77.0ms
(* (/ t l) (sin k))

simplify22.0s

Counts
112 → 131
Calls

112 calls. Slowest were:

846.0ms
(* (* (* t (* t (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
807.0ms
(* (/ l t) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
760.0ms
(* (* (* (* (* (/ t (/ l t)) (/ t (/ l t))) (/ t (/ l t))) (* (* (* (/ t l) (sin k)) (* (/ t l) (sin k))) (* (/ t l) (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)))

prune2.2s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 7.0b

localize31.0ms

Local error

Found 4 expressions with local error:

9.6b
(* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1))
7.3b
(/ (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (/ l t) (cos k)))
4.0b
(* (/ t l) (sin k))
3.8b
(* (* t (* (/ t l) (sin k))) (sin k))

rewrite99.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

51.0ms
(/ (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (/ l t) (cos k)))
32.0ms
(* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1))
9.0ms
(* (* t (* (/ t l) (sin k))) (sin k))

series658.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

318.0ms
(/ (* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1)) (* (/ l t) (cos k)))
204.0ms
(* (* (* t (* (/ t l) (sin k))) (sin k)) (+ (+ 1 (pow (/ k t) 2)) 1))
75.0ms
(* (* t (* (/ t l) (sin k))) (sin k))
61.0ms
(* (/ t l) (sin k))

simplify14.2s

Counts
68 → 103
Calls

68 calls. Slowest were:

983.0ms
(/ (* (* (* (* (* 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))))
832.0ms
(/ (* (* (* (* (* 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))))
789.0ms
(/ (* (* (* (* (* 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))))

prune1.7s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 4.8b

regimes275.0ms

Accuracy

43.1% (5.8b remaining)

Error of 11.4b against oracle of 5.6b and baseline of 15.8b

bsearch770.0ms

end0.0ms

sample15.7s

Algorithm
intervals