Average Error: 12.2 → 8.8
Time: 37.4s
Precision: 64
Internal Precision: 128
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
↓
\[\frac{\sin th}{\frac{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}{\sin ky}}\]

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 12.2

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

  2. Initial simplification11.2

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

  4. Applied associate-/l*8.8

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

  5. Taylor expanded around inf 12.3

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

  6. Simplified8.8

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

  7. Final simplification8.8

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

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(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: 20.3s)Debug log

start865.0ms

Algorithm
intervals

setup66.0ms

Pruning

2 alts after pruning (2 fresh and 0 done)

Merged error: 9.0b

localize26.0ms

Local error

Found 3 expressions with local error:

2.2b
(/ (* (sin th) (sin ky)) (hypot (sin kx) (sin ky)))
0.2b
(* (sin th) (sin ky))
0.1b
(hypot (sin kx) (sin ky))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
3 → 54
Calls

3 calls. Slowest were:

8.0ms
(/ (* (sin th) (sin ky)) (hypot (sin kx) (sin ky)))
4.0ms
(* (sin th) (sin ky))
0.0ms
(hypot (sin kx) (sin ky))

series518.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

310.0ms
(/ (* (sin th) (sin ky)) (hypot (sin kx) (sin ky)))
131.0ms
(hypot (sin kx) (sin ky))
77.0ms
(* (sin th) (sin ky))

simplify1.7s

Counts
33 → 63
Calls

33 calls. Slowest were:

599.0ms
(/ (* (* (* (sin th) (sin ky)) (* (sin th) (sin ky))) (* (sin th) (sin ky))) (* (* (hypot (sin kx) (sin ky)) (hypot (sin kx) (sin ky))) (hypot (sin kx) (sin ky))))
485.0ms
(/ (* (* (* (sin th) (sin th)) (sin th)) (* (* (sin ky) (sin ky)) (sin ky))) (* (* (hypot (sin kx) (sin ky)) (hypot (sin kx) (sin ky))) (hypot (sin kx) (sin ky))))
328.0ms
(- (+ (* 1/12 (* (pow kx 2) ky)) ky) (* 1/6 (pow ky 3)))

prune871.0ms

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 7.5b

localize7.0ms

Local error

Found 3 expressions with local error:

0.2b
(* (sin th) (/ (sin ky) (hypot (sin kx) (sin ky))))
0.1b
(/ (sin ky) (hypot (sin kx) (sin ky)))
0.1b
(hypot (sin kx) (sin ky))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 71
Calls

3 calls. Slowest were:

6.0ms
(* (sin th) (/ (sin ky) (hypot (sin kx) (sin ky))))
3.0ms
(/ (sin ky) (hypot (sin kx) (sin ky)))
0.0ms
(hypot (sin kx) (sin ky))

series592.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

261.0ms
(* (sin th) (/ (sin ky) (hypot (sin kx) (sin ky))))
209.0ms
(/ (sin ky) (hypot (sin kx) (sin ky)))
122.0ms
(hypot (sin kx) (sin ky))

simplify1.8s

Counts
55 → 80
Calls

55 calls. Slowest were:

509.0ms
(* (* (* (sin th) (sin th)) (sin th)) (/ (* (* (sin ky) (sin ky)) (sin ky)) (* (* (hypot (sin kx) (sin ky)) (hypot (sin kx) (sin ky))) (hypot (sin kx) (sin ky)))))
362.0ms
(- (+ (* 1/12 (* (pow kx 2) ky)) ky) (* 1/6 (pow ky 3)))
210.0ms
(- (+ (* 1/6 (* kx ky)) (* 7/360 (* (pow kx 3) ky))) (* 71/720 (* kx (pow ky 3))))

prune1.2s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 7.5b

localize9.0ms

Local error

Found 3 expressions with local error:

0.2b
(/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky)))
0.1b
(/ (hypot (sin kx) (sin ky)) (sin ky))
0.1b
(hypot (sin kx) (sin ky))

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
3 → 111
Calls

3 calls. Slowest were:

5.0ms
(/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky)))
3.0ms
(/ (hypot (sin kx) (sin ky)) (sin ky))
0.0ms
(hypot (sin kx) (sin ky))

series552.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

228.0ms
(/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky)))
190.0ms
(/ (hypot (sin kx) (sin ky)) (sin ky))
134.0ms
(hypot (sin kx) (sin ky))

simplify3.5s

Counts
132 → 120
Calls

132 calls. Slowest were:

452.0ms
(/ (* (* (sin th) (sin th)) (sin th)) (/ (* (* (hypot (sin kx) (sin ky)) (hypot (sin kx) (sin ky))) (hypot (sin kx) (sin ky))) (* (* (sin ky) (sin ky)) (sin ky))))
452.0ms
(/ (* (cbrt (sin th)) (cbrt (sin th))) (/ (* (cbrt (hypot (sin kx) (sin ky))) (cbrt (hypot (sin kx) (sin ky)))) (* (cbrt (sin ky)) (cbrt (sin ky)))))
340.0ms
(- (+ (* 1/12 (* (pow kx 2) ky)) ky) (* 1/6 (pow ky 3)))

prune1.7s

Pruning

19 alts after pruning (19 fresh and 0 done)

Merged error: 7.5b

localize1.0ms

Local error

Found 3 expressions with local error:

0.2b
(/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky)))
0.1b
(/ (hypot (sin kx) (sin ky)) (sin ky))
0.1b
(hypot (sin kx) (sin ky))

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
3 → 111
Calls

3 calls. Slowest were:

6.0ms
(/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky)))
4.0ms
(/ (hypot (sin kx) (sin ky)) (sin ky))
0.0ms
(hypot (sin kx) (sin ky))

series604.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

266.0ms
(/ (sin th) (/ (hypot (sin kx) (sin ky)) (sin ky)))
190.0ms
(/ (hypot (sin kx) (sin ky)) (sin ky))
148.0ms
(hypot (sin kx) (sin ky))

simplify3.8s

Counts
132 → 120
Calls

132 calls. Slowest were:

499.0ms
(/ (* (cbrt (sin th)) (cbrt (sin th))) (/ (* (cbrt (hypot (sin kx) (sin ky))) (cbrt (hypot (sin kx) (sin ky)))) (* (cbrt (sin ky)) (cbrt (sin ky)))))
443.0ms
(/ (* (* (sin th) (sin th)) (sin th)) (/ (* (* (hypot (sin kx) (sin ky)) (hypot (sin kx) (sin ky))) (hypot (sin kx) (sin ky))) (* (* (sin ky) (sin ky)) (sin ky))))
358.0ms
(- (+ (* 1/12 (* (pow kx 2) ky)) ky) (* 1/6 (pow ky 3)))

prune1.5s

Pruning

19 alts after pruning (18 fresh and 1 done)

Merged error: 7.5b

regimes952.0ms

Accuracy

0% (0.1b remaining)

Error of 8.8b against oracle of 8.6b and baseline of 8.8b

bsearch1.0ms