Average Error: 15.2 → 0.4
Time: 28.7s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}\right)}^{3}}} \cdot r\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.2

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  2. Using strategy rm

  3. Applied cos-sum0.3

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
  4. Using strategy rm

  5. Applied add-cbrt-cube0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \color{blue}{\sqrt[3]{\left(\sin b \cdot \sin b\right) \cdot \sin b}}}\]

  6. Applied add-cbrt-cube0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\sin a \cdot \sin a\right) \cdot \sin a}} \cdot \sqrt[3]{\left(\sin b \cdot \sin b\right) \cdot \sin b}}\]

  7. Applied cbrt-unprod0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sqrt[3]{\left(\left(\sin a \cdot \sin a\right) \cdot \sin a\right) \cdot \left(\left(\sin b \cdot \sin b\right) \cdot \sin b\right)}}}\]

  8. Simplified0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{\color{blue}{{\left(\sin b \cdot \sin a\right)}^{3}}}}\]
  9. Using strategy rm

  10. Applied add-cbrt-cube0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sin b \cdot \color{blue}{\sqrt[3]{\left(\sin a \cdot \sin a\right) \cdot \sin a}}\right)}^{3}}}\]

  11. Applied add-cbrt-cube0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\color{blue}{\sqrt[3]{\left(\sin b \cdot \sin b\right) \cdot \sin b}} \cdot \sqrt[3]{\left(\sin a \cdot \sin a\right) \cdot \sin a}\right)}^{3}}}\]

  12. Applied cbrt-unprod0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\color{blue}{\left(\sqrt[3]{\left(\left(\sin b \cdot \sin b\right) \cdot \sin b\right) \cdot \left(\left(\sin a \cdot \sin a\right) \cdot \sin a\right)}\right)}}^{3}}}\]

  13. Simplified0.4

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left(\sin a \cdot \sin b\right)}^{3}}}\right)}^{3}}}\]

  14. Final simplification0.4

    \[\leadsto \frac{\sin b}{\cos a \cdot \cos b - \sqrt[3]{{\left(\sqrt[3]{{\left(\sin a \cdot \sin b\right)}^{3}}\right)}^{3}}} \cdot r\]

Reproduce

herbie shell --seed 2019005 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))

Details

Time bar (total: 27.4s)Debug log

sample760.0ms

Algorithm
intervals

simplify49.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

49.0ms
(* r (/ (sin b) (cos (+ a b))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.2b

localize33.0ms

Local error

Found 3 expressions with local error:

5.7b
(cos (+ a b))
0.2b
(/ (sin b) (cos (+ a b)))
0.2b
(* r (/ (sin b) (cos (+ a b))))

rewrite21.0ms

Algorithm
rewrite-expression-head
Counts
3 → 57
Calls

3 calls. Slowest were:

7.0ms
(/ (sin b) (cos (+ a b)))
6.0ms
(cos (+ a b))
5.0ms
(* r (/ (sin b) (cos (+ a b))))

series202.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

114.0ms
(* r (/ (sin b) (cos (+ a b))))
61.0ms
(/ (sin b) (cos (+ a b)))
26.0ms
(cos (+ a b))

simplify787.0ms

Counts
49 → 66
Calls

49 calls. Slowest were:

180.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
137.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
110.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))

prune704.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.2b

localize32.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
0.2b
(* (sin a) (sin b))
0.2b
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
0.1b
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

rewrite32.0ms

Algorithm
rewrite-expression-head
Counts
4 → 89
Calls

4 calls. Slowest were:

11.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
10.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
7.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

series540.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

241.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
150.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
87.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
61.0ms
(* (sin a) (sin b))

simplify2.4s

Counts
70 → 101
Calls

70 calls. Slowest were:

392.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
377.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
203.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))

prune1.5s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

26.8b
(cbrt (pow (* (sin b) (sin a)) 3))
0.3b
(pow (* (sin b) (sin a)) 3)
0.2b
(/ (sin b) (- (* (cos a) (cos b)) (cbrt (pow (* (sin b) (sin a)) 3))))
0.2b
(* (sin b) (sin a))

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 84
Calls

4 calls. Slowest were:

23.0ms
(/ (sin b) (- (* (cos a) (cos b)) (cbrt (pow (* (sin b) (sin a)) 3))))
7.0ms
(* (sin b) (sin a))
3.0ms
(pow (* (sin b) (sin a)) 3)

series258.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

98.0ms
(/ (sin b) (- (* (cos a) (cos b)) (cbrt (pow (* (sin b) (sin a)) 3))))
58.0ms
(pow (* (sin b) (sin a)) 3)
58.0ms
(cbrt (pow (* (sin b) (sin a)) 3))
43.0ms
(* (sin b) (sin a))

simplify3.4s

Counts
67 → 96
Calls

67 calls. Slowest were:

253.0ms
(pow (- (cos (- b a)) (cos (+ b a))) 3)
251.0ms
(cbrt (pow (- (cos (- b a)) (cos (+ b a))) 3))
200.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (- (* (cos a) (cos b)) (cbrt (pow (* (sin b) (sin a)) 3))) (- (* (cos a) (cos b)) (cbrt (pow (* (sin b) (sin a)) 3)))) (- (* (cos a) (cos b)) (cbrt (pow (* (sin b) (sin a)) 3)))))

prune1.6s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

26.8b
(cbrt (pow (cbrt (pow (* (sin a) (sin b)) 3)) 3))
26.8b
(cbrt (pow (* (sin a) (sin b)) 3))
0.3b
(pow (cbrt (pow (* (sin a) (sin b)) 3)) 3)
0.3b
(pow (* (sin a) (sin b)) 3)

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 106
Calls

4 calls. Slowest were:

3.0ms
(pow (* (sin a) (sin b)) 3)
2.0ms
(cbrt (pow (cbrt (pow (* (sin a) (sin b)) 3)) 3))
2.0ms
(pow (cbrt (pow (* (sin a) (sin b)) 3)) 3)

series274.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

85.0ms
(pow (* (sin a) (sin b)) 3)
65.0ms
(cbrt (pow (cbrt (pow (* (sin a) (sin b)) 3)) 3))
63.0ms
(pow (cbrt (pow (* (sin a) (sin b)) 3)) 3)
60.0ms
(cbrt (pow (* (sin a) (sin b)) 3))

simplify4.7s

Counts
82 → 118
Calls

82 calls. Slowest were:

285.0ms
(pow (cbrt (pow (- (cos (- a b)) (cos (+ a b))) 3)) 3)
259.0ms
(pow (* (cbrt (cbrt (pow (* (sin a) (sin b)) 3))) (cbrt (cbrt (pow (* (sin a) (sin b)) 3)))) 3)
257.0ms
(cbrt (pow (* (cbrt (cbrt (pow (* (sin a) (sin b)) 3))) (cbrt (cbrt (pow (* (sin a) (sin b)) 3)))) 3))

prune2.1s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

regimes364.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.1b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample7.6s

Algorithm
intervals