Average Error: 12.4 → 12.7
Time: 37.5s
Precision: 64
Internal Precision: 128
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\left(\left(\sqrt{\frac{1}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}}} \cdot \sin th\right) \cdot \sqrt{\frac{1}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}}}\right) \cdot \sin ky\]

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.4

    \[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
  2. Using strategy rm

  3. Applied div-inv12.4

    \[\leadsto \color{blue}{\left(\sin ky \cdot \frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}\right)} \cdot \sin th\]

  4. Applied associate-*l*12.5

    \[\leadsto \color{blue}{\sin ky \cdot \left(\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\right)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt12.7

    \[\leadsto \sin ky \cdot \left(\color{blue}{\left(\sqrt{\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sqrt{\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}}\right)} \cdot \sin th\right)\]

  7. Applied associate-*l*12.7

    \[\leadsto \sin ky \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}} \cdot \sin th\right)\right)}\]

  8. Final simplification12.7

    \[\leadsto \left(\left(\sqrt{\frac{1}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}}} \cdot \sin th\right) \cdot \sqrt{\frac{1}{\sqrt{{\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}}}}\right) \cdot \sin ky\]

Reproduce

herbie shell --seed 2019008 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))

Details

Time bar (total: 33.6s)Debug log

sample581.0ms

Algorithm
intervals

simplify23.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

22.0ms
(* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 11.7b

localize70.0ms

Local error

Found 4 expressions with local error:

2.4b
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.3b
(pow (sin ky) 2)
0.2b
(pow (sin kx) 2)
0.2b
(* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))

rewrite46.0ms

Algorithm
rewrite-expression-head
Counts
4 → 91
Calls

4 calls. Slowest were:

23.0ms
(* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))
15.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
2.0ms
(pow (sin ky) 2)

series312.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

155.0ms
(* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))
119.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
21.0ms
(pow (sin ky) 2)
17.0ms
(pow (sin kx) 2)

simplify1.6s

Counts
68 → 103
Calls

68 calls. Slowest were:

467.0ms
(* (/ (* (* (sin ky) (sin ky)) (sin ky)) (* (* (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (* (sin th) (sin th)) (sin th)))
210.0ms
(* (* (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (* (sin th) (sin th)) (sin th)))
131.0ms
(sqrt (- (* (pow (sin kx) 2) (pow (sin kx) 2)) (* (pow (sin ky) 2) (pow (sin ky) 2))))

prune1.7s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 10.9b

localize19.0ms

Local error

Found 4 expressions with local error:

2.4b
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.3b
(* (sin ky) (* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))
0.3b
(* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))
0.3b
(pow (sin ky) 2)

rewrite49.0ms

Algorithm
rewrite-expression-head
Counts
4 → 72
Calls

4 calls. Slowest were:

21.0ms
(* (sin ky) (* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))
14.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
10.0ms
(* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))

series433.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

172.0ms
(* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))
165.0ms
(* (sin ky) (* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))
80.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
16.0ms
(pow (sin ky) 2)

simplify2.5s

Counts
48 → 84
Calls

48 calls. Slowest were:

461.0ms
(* (* (* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (* (sin th) (sin th)) (sin th)))
441.0ms
(* (* (* (sin ky) (sin ky)) (sin ky)) (* (* (* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)) (* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))) (* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th))))
378.0ms
(* (* (* (sin ky) (sin ky)) (sin ky)) (* (* (* (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (* (sin th) (sin th)) (sin th))))

prune1.3s

Pruning

20 alts after pruning (20 fresh and 0 done)

Merged error: 10.9b

localize17.0ms

Local error

Found 4 expressions with local error:

2.4b
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
2.4b
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.4b
(* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th)))
0.3b
(* (sin ky) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th))))

rewrite71.0ms

Algorithm
rewrite-expression-head
Counts
4 → 83
Calls

4 calls. Slowest were:

40.0ms
(* (sin ky) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th))))
15.0ms
(* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th)))
7.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))

series477.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

151.0ms
(* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th)))
130.0ms
(* (sin ky) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th))))
108.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
89.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))

simplify3.3s

Counts
62 → 95
Calls

62 calls. Slowest were:

527.0ms
(* (* (* (sin ky) (sin ky)) (sin ky)) (* (* (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th)))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (* (sqrt (/ 1 (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))) (sin th)))))
227.0ms
(- (+ (* 1/12 (* (pow kx 2) ky)) ky) (* 1/6 (pow ky 3)))
217.0ms
(- (+ (* 1/12 (* (pow kx 2) ky)) ky) (* 1/6 (pow ky 3)))

prune1.9s

Pruning

21 alts after pruning (20 fresh and 1 done)

Merged error: 10.9b

localize31.0ms

Local error

Found 4 expressions with local error:

2.4b
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
2.4b
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
2.4b
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
0.4b
(cbrt (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))))

rewrite55.0ms

Algorithm
rewrite-expression-head
Counts
4 → 87
Calls

4 calls. Slowest were:

14.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
14.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
14.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))

series472.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

224.0ms
(cbrt (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))))
96.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
83.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))
68.0ms
(sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))

simplify1.8s

Counts
67 → 99
Calls

67 calls. Slowest were:

193.0ms
(- (+ (* 1/12 (* (pow kx 2) ky)) ky) (* 1/6 (pow ky 3)))
177.0ms
(- (+ (* 1/12 (* (pow kx 2) ky)) ky) (* 1/6 (pow ky 3)))
170.0ms
(sqrt (- (* (pow (sin kx) 2) (pow (sin kx) 2)) (* (pow (sin ky) 2) (pow (sin ky) 2))))

prune1.9s

Pruning

23 alts after pruning (22 fresh and 1 done)

Merged error: 10.9b

regimes2.5s

Accuracy

0% (1.6b remaining)

Error of 12.7b against oracle of 11.1b and baseline of 12.7b

bsearch4.0ms

end0.0ms

sample12.3s

Algorithm
intervals