Average Error: 31.6 → 14.8
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -8.522503082318411 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k}} \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{elif}\;\ell \le -1.9796816178314917 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} + \frac{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;\ell \le 6.253093589310313 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{t}{\ell} \cdot \left(k - \left(\frac{1}{6} \cdot k\right) \cdot \left(k \cdot k\right)\right)\right) \cdot t\right) \cdot \tan k}{\frac{\ell}{t}} \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k}} \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \end{array}\]

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 3 regimes

  2. if l < -8.522503082318411e+128 or 6.253093589310313e-29 < l

    1. Initial program 47.1

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

    3. Applied unpow347.1

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

    4. Applied times-frac36.0

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

    5. Applied associate-*l*35.7

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

    7. Applied associate-/l*24.6

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

    9. Applied associate-*l/22.6

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

    10. Applied associate-*l/20.9

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

    12. Applied clear-num21.0

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

    if -8.522503082318411e+128 < l < -1.9796816178314917e-115

    1. Initial program 25.9

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

    3. Applied unpow325.9

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

    4. Applied times-frac23.1

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

    5. Applied associate-*l*21.6

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

    6. Taylor expanded around -inf 18.5

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

    if -1.9796816178314917e-115 < l < 6.253093589310313e-29

    1. Initial program 23.5

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

    3. Applied unpow323.5

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

    4. Applied times-frac18.0

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

    5. Applied associate-*l*15.1

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

    7. Applied associate-/l*11.0

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

    9. Applied associate-*l/10.8

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

    10. Applied associate-*l/10.3

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

    11. Taylor expanded around 0 30.2

      \[\leadsto \frac{2}{\frac{\left(t \cdot \color{blue}{\left(\frac{t \cdot k}{\ell} - \frac{1}{6} \cdot \frac{t \cdot {k}^{3}}{\ell}\right)}\right) \cdot \tan k}{\frac{\ell}{t}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

    12. Simplified8.7

      \[\leadsto \frac{2}{\frac{\left(t \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \left(k - \left(\frac{1}{6} \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)}\right) \cdot \tan k}{\frac{\ell}{t}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -8.522503082318411 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k}} \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{elif}\;\ell \le -1.9796816178314917 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} + \frac{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;\ell \le 6.253093589310313 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{t}{\ell} \cdot \left(k - \left(\frac{1}{6} \cdot k\right) \cdot \left(k \cdot k\right)\right)\right) \cdot t\right) \cdot \tan k}{\frac{\ell}{t}} \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k}} \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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

sample442.0ms

Algorithm
intervals

simplify154.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

154.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1)))

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.9b

localize59.0ms

Local error

Found 4 expressions with local error:

15.0b
(/ (pow t 3) (* l l))
12.4b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
4.3b
(* (/ (pow t 3) (* l l)) (sin k))
2.3b
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))

rewrite78.0ms

Algorithm
rewrite-expression-head
Counts
4 → 112
Calls

4 calls. Slowest were:

42.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
19.0ms
(* (* (/ (pow t 3) (* l l)) (sin k)) (tan k))
10.0ms
(* (/ (pow t 3) (* l l)) (sin k))

series597.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

353.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
150.0ms
(* (* (/ (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))

simplify15.9s

Counts
98 → 124
Calls

98 calls. Slowest were:

978.0ms
(* (* (* (/ (* (* (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)))
728.0ms
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
700.0ms
(* (* (* (pow t 3) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))

prune2.3s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 13.7b

localize44.0ms

Local error

Found 4 expressions with local error:

12.4b
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
7.0b
(/ (* t t) l)
3.1b
(* (/ t l) (sin k))
2.3b
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))

rewrite167.0ms

Algorithm
rewrite-expression-head
Counts
4 → 107
Calls

4 calls. Slowest were:

130.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
27.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
4.0ms
(* (/ t l) (sin k))

series591.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

317.0ms
(* (* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
181.0ms
(* (* (/ (* t t) l) (* (/ t l) (sin k))) (tan k))
83.0ms
(* (/ t l) (sin k))
10.0ms
(/ (* t t) l)

simplify21.6s

Counts
99 → 119
Calls

99 calls. Slowest were:

958.0ms
(* (* l (cos k)) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
736.0ms
(* (* l l) (- (+ 1 (pow (/ k t) 2)) 1))
684.0ms
(* (cos k) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))

prune1.9s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 8.6b

localize34.0ms

Local error

Found 4 expressions with local error:

12.4b
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
3.1b
(* (/ t l) (sin k))
2.3b
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
1.1b
(* (/ t (/ l t)) (* (/ t l) (sin k)))

rewrite182.0ms

Algorithm
rewrite-expression-head
Counts
4 → 119
Calls

4 calls. Slowest were:

121.0ms
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
34.0ms
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
19.0ms
(* (/ t (/ l t)) (* (/ t l) (sin k)))

series675.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

346.0ms
(* (* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))
176.0ms
(* (* (/ t (/ l t)) (* (/ t l) (sin k))) (tan k))
90.0ms
(* (/ t l) (sin k))
63.0ms
(* (/ t (/ l t)) (* (/ t l) (sin k)))

simplify24.2s

Counts
112 → 131
Calls

112 calls. Slowest were:

982.0ms
(* (* l (cos k)) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
826.0ms
(* (/ l t) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))
747.0ms
(* (/ l t) (- (+ 1 (pow (/ k t) 2)) 1))

prune2.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 6.6b

localize29.0ms

Local error

Found 4 expressions with local error:

12.4b
(* (/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t)) (+ (+ 1 (pow (/ k t) 2)) 1))
3.5b
(* (* t (* (/ t l) (sin k))) (tan k))
3.1b
(* (/ t l) (sin k))
1.5b
(/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t))

rewrite50.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

19.0ms
(* (/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t)) (+ (+ 1 (pow (/ k t) 2)) 1))
13.0ms
(* (* t (* (/ t l) (sin k))) (tan k))
13.0ms
(/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t))

series781.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

331.0ms
(* (/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t)) (+ (+ 1 (pow (/ k t) 2)) 1))
198.0ms
(* (* t (* (/ t l) (sin k))) (tan k))
182.0ms
(/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t))
70.0ms
(* (/ t l) (sin k))

simplify13.2s

Counts
78 → 110
Calls

78 calls. Slowest were:

798.0ms
(* (/ l t) (- (+ 1 (pow (/ k t) 2)) 1))
686.0ms
(* (* (* (/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t)) (/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t))) (/ (* (* t (* (/ t l) (sin k))) (tan k)) (/ l t))) (* (* (+ (+ 1 (pow (/ k t) 2)) 1) (+ (+ 1 (pow (/ k t) 2)) 1)) (+ (+ 1 (pow (/ k t) 2)) 1)))
685.0ms
(* (/ l t) (+ (* (+ 1 (pow (/ k t) 2)) (+ 1 (pow (/ k t) 2))) (- (* 1 1) (* (+ 1 (pow (/ k t) 2)) 1))))

prune1.9s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 6.6b

regimes356.0ms

Accuracy

8.3% (6.8b remaining)

Error of 14.8b against oracle of 8.0b and baseline of 15.4b

bsearch937.0ms

end0.0ms

sample15.3s

Algorithm
intervals