Average Error: 47.5 → 16.8
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\left(\frac{\sqrt[3]{\cos k}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \ell\right)\right) \cdot 2}{\frac{{k}^{2}}{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}}\]

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. Initial program 47.5

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

  2. Taylor expanded around inf 23.5

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

  4. Applied times-frac23.5

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

  5. Applied associate-/r*23.4

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

  7. Applied add-cube-cbrt23.5

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

  8. Applied times-frac23.5

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

  9. Applied div-inv23.5

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

  10. Applied times-frac22.3

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

  11. Simplified16.9

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

  13. Applied associate-*l/16.8

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

  14. Final simplification16.8

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

Reproduce

herbie shell --seed 2019010 
(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

sample1.5s

Algorithm
intervals

simplify131.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
131.0ms
(/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1)))

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 45.1b

localize72.0ms

Local error

Found 4 expressions with local error:

28.8b
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
16.1b
(/ (pow t 3) (* l l))
1.4b
(- (+ 1 (pow (/ k t) 2)) 1)
1.3b
(* (/ (pow t 3) (* l l)) (sin k))

rewrite62.0ms

Algorithm
rewrite-expression-head
Rules
32×add-exp-log
25×add-cbrt-cube
15×prod-exp
15×pow1
13×associate-*l/
12×add-sqr-sqrt
11×cbrt-unprod
10×add-cube-cbrt
10×associate-*l*
10×times-frac
10×*-un-lft-identity
9×frac-times
7×pow-prod-down
7×associate-*r*
6×tan-quot
6×div-exp
5×associate-/l*
5×flip--
5×associate-*r/
5×flip3--
4×add-log-exp
3×associate--l+
3×unpow3
3×distribute-lft-in
3×sub-neg
3×cbrt-undiv
3×pow-to-exp
3×cube-mult
3×distribute-rgt-in
2×div-inv
2×*-commutative
2×difference-of-sqr-1
1×frac-2neg
1×associate-/r*
1×clear-num
Counts
4 → 106
Calls
4 calls:
Slowest
36.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
13.0ms
(* (/ (pow t 3) (* l l)) (sin k))
7.0ms
(- (+ 1 (pow (/ k t) 2)) 1)
3.0ms
(/ (pow t 3) (* l l))

series493.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
387.0ms
(* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
72.0ms
(* (/ (pow t 3) (* l l)) (sin k))
18.0ms
(/ (pow t 3) (* l l))
16.0ms
(- (+ 1 (pow (/ k t) 2)) 1)

simplify16.2s

Counts
87 → 118
Calls
87 calls:
Slowest
880.0ms
(* (* (* (pow t 3) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))
811.0ms
(* (* (* (pow t 3) (sin k)) (sin k)) (- (+ 1 (pow (/ k t) 2)) 1))
614.0ms
(sqrt (- (+ 1 (pow (/ k t) 2)) 1))
594.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
591.0ms
(* (* (* (pow t 3) (sin k)) (sin k)) (- (pow (+ 1 (pow (/ k t) 2)) 3) (pow 1 3)))

prune1.9s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 15.5b

localize48.0ms

Local error

Found 4 expressions with local error:

17.1b
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
6.8b
(* t (* (pow k 2) (pow (sin k) 2)))
0.7b
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
0.4b
(* (pow k 2) (pow (sin k) 2))

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
18×add-exp-log
11×add-cbrt-cube
8×add-cube-cbrt
8×prod-exp
8×associate-*r*
8×*-un-lft-identity
8×add-sqr-sqrt
6×associate-/r*
6×pow-to-exp
4×add-log-exp
4×associate-*l*
4×pow1
3×div-inv
3×div-exp
3×unpow-prod-down
2×unpow2
2×cbrt-unprod
2×*-commutative
2×pow-exp
2×times-frac
2×frac-2neg
2×clear-num
2×cbrt-undiv
1×associate-/l*
1×associate-/r/
1×pow-prod-down
Counts
4 → 70
Calls
4 calls:
Slowest
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)))
9.0ms
(* (pow k 2) (pow (sin k) 2))
2.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))

series627.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
269.0ms
(/ 2 (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
233.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
98.0ms
(* t (* (pow k 2) (pow (sin k) 2)))
27.0ms
(* (pow k 2) (pow (sin k) 2))

simplify6.1s

Counts
44 → 82
Calls
44 calls:
Slowest
583.0ms
(/ (* (* (* 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))))
562.0ms
(- (* 2 (/ (pow l 2) (* t (pow k 4)))) (* 1/3 (/ (pow l 2) (* t (pow k 2)))))
549.0ms
(sqrt (/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k))))
545.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))
500.0ms
(/ (* t (* (pow k 2) (pow (sin k) 2))) (* (pow l 2) (cos k)))

prune1.3s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 11.9b

localize10.0ms

Local error

Found 4 expressions with local error:

14.3b
(/ (/ 2 (/ t (pow l 2))) (/ (* (pow k 2) (pow (sin k) 2)) (cos k)))
6.9b
(/ t (pow l 2))
0.7b
(/ 2 (/ t (pow l 2)))
0.4b
(* (pow k 2) (pow (sin k) 2))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
53×times-frac
35×add-exp-log
29×add-cube-cbrt
29×*-un-lft-identity
29×add-sqr-sqrt
18×div-inv
15×div-exp
15×add-cbrt-cube
15×associate-/r*
12×prod-exp
10×associate-/r/
10×pow-to-exp
7×associate-*r*
5×associate-/l*
5×cbrt-undiv
4×add-log-exp
4×pow-exp
4×associate-*l*
4×pow1
3×unpow2
3×frac-2neg
3×unpow-prod-down
3×clear-num
2×cbrt-unprod
1×associate-/l/
1×*-commutative
1×pow-prod-down
Counts
4 → 126
Calls
4 calls:
Slowest
6.0ms
(* (pow k 2) (pow (sin k) 2))
5.0ms
(/ (/ 2 (/ t (pow l 2))) (/ (* (pow k 2) (pow (sin k) 2)) (cos k)))
0.0ms
(/ t (pow l 2))
0.0ms
(/ 2 (/ t (pow l 2)))

series335.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
254.0ms
(/ (/ 2 (/ t (pow l 2))) (/ (* (pow k 2) (pow (sin k) 2)) (cos k)))
47.0ms
(* (pow k 2) (pow (sin k) 2))
21.0ms
(/ 2 (/ t (pow l 2)))
13.0ms
(/ t (pow l 2))

simplify12.1s

Counts
129 → 138
Calls
129 calls:
Slowest
510.0ms
(/ (/ 2 (/ t (pow l 2))) (/ (* (pow k 2) (pow (sin k) 2)) (cos k)))
438.0ms
(/ (/ 1 (/ t (pow l 2))) (sqrt (/ (* (pow k 2) (pow (sin k) 2)) (cos k))))
422.0ms
(/ (* (cbrt (/ 2 (/ t (pow l 2)))) (cbrt (/ 2 (/ t (pow l 2))))) (* (pow k 2) (pow (sin k) 2)))
419.0ms
(/ (* (cbrt (/ 2 (/ t (pow l 2)))) (cbrt (/ 2 (/ t (pow l 2))))) (/ (pow k 2) 1))
400.0ms
(/ (/ 1 (/ t (pow l 2))) (/ (* (pow k 2) (pow (sin k) 2)) (cos k)))

prune2.2s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 7.6b

localize26.0ms

Local error

Found 4 expressions with local error:

10.4b
(* (/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k))))) (* (/ (cbrt (cos k)) (sin k)) (* (/ (/ l t) (sin k)) l)))
2.7b
(* (/ (cbrt (cos k)) (sin k)) (* (/ (/ l t) (sin k)) l))
2.7b
(/ (/ l t) (sin k))
0.5b
(/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k)))))

rewrite64.0ms

Algorithm
rewrite-expression-head
Rules
34×times-frac
29×add-cube-cbrt
29×*-un-lft-identity
29×add-sqr-sqrt
27×associate-*l*
20×add-exp-log
18×add-cbrt-cube
11×associate-/r*
11×pow1
10×associate-*l/
9×div-inv
9×cbrt-prod
7×cbrt-unprod
7×prod-exp
6×associate-*r/
6×frac-times
4×add-log-exp
4×associate-/l*
4×pow-prod-down
3×div-exp
3×cbrt-undiv
2×*-commutative
2×associate-/r/
2×associate-*r*
2×frac-2neg
2×clear-num
1×associate-/l/
1×unpow2
Counts
4 → 119
Calls
4 calls:
Slowest
45.0ms
(* (/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k))))) (* (/ (cbrt (cos k)) (sin k)) (* (/ (/ l t) (sin k)) l)))
12.0ms
(* (/ (cbrt (cos k)) (sin k)) (* (/ (/ l t) (sin k)) l))
3.0ms
(/ (/ l t) (sin k))
2.0ms
(/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k)))))

series692.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
259.0ms
(/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k)))))
203.0ms
(* (/ (cbrt (cos k)) (sin k)) (* (/ (/ l t) (sin k)) l))
146.0ms
(* (/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k))))) (* (/ (cbrt (cos k)) (sin k)) (* (/ (/ l t) (sin k)) l)))
84.0ms
(/ (/ l t) (sin k))

simplify7.8s

Counts
104 → 131
Calls
104 calls:
Slowest
412.0ms
(+ (log (/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k)))))) (+ (- (log (cbrt (cos k))) (log (sin k))) (log (* (/ (/ l t) (sin k)) l))))
408.0ms
(* (* (* (/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k))))) (/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k)))))) (/ 2 (/ (pow k 2) (* (cbrt (cos k)) (cbrt (cos k)))))) (* (/ (cos k) (* (* (sin k) (sin k)) (sin k))) (* (* (* (/ (/ l t) (sin k)) l) (* (/ (/ l t) (sin k)) l)) (* (/ (/ l t) (sin k)) l))))
403.0ms
(* 2 (* (cbrt (cos k)) (* (/ (/ l t) (sin k)) l)))
403.0ms
(* 2 (* (/ 1 (pow k 2)) (pow (pow (cos k) 2) 1/3)))
378.0ms
(+ (- (log (cbrt (cos k))) (log (sin k))) (log (* (/ (/ l t) (sin k)) l)))

prune1.9s

Pruning

15 alts after pruning (15 fresh and 0 done)

Merged error: 7.3b

regimes388.0ms

Accuracy

0% (7.2b remaining)

Error of 16.8b against oracle of 9.6b and baseline of 16.8b

bsearch4.0ms

end0.0ms

sample41.6s

Algorithm
intervals