Average Error: 14.7 → 0.4
Time: 24.1s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sqrt[3]{\left(\sin b \cdot \sin a\right) \cdot \left(\left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right)}}\]

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 14.7

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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-log-exp0.4

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

  7. Applied associate-*r/0.4

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

  8. Simplified0.3

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

  10. Applied add-cbrt-cube0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019010 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))

Details

Time bar (total: 22.1s)Debug log

sample414.0ms

Algorithm
intervals

simplify47.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
46.0ms
(* r (/ (sin b) (cos (+ a b))))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.9b

localize40.0ms

Local error

Found 3 expressions with local error:

2.1b
(cos (+ a b))
0.2b
(/ (sin b) (cos (+ a b)))
0.2b
(* r (/ (sin b) (cos (+ a b))))

rewrite23.0ms

Algorithm
rewrite-expression-head
Rules
18×add-cube-cbrt
18×times-frac
18×*-un-lft-identity
18×add-sqr-sqrt
13×associate-*r*
5×add-exp-log
5×add-cbrt-cube
3×add-log-exp
3×associate-/l*
3×associate-/r*
3×pow1
2×div-inv
1×*-commutative
1×associate-*r/
1×div-exp
1×frac-2neg
1×clear-num
1×cos-sum
1×cbrt-undiv
Counts
3 → 57
Calls
3 calls:
Slowest
8.0ms
(/ (sin b) (cos (+ a b)))
7.0ms
(cos (+ a b))
5.0ms
(* r (/ (sin b) (cos (+ a b))))

series401.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
146.0ms
(* r (/ (sin b) (cos (+ a b))))
136.0ms
(cos (+ a b))
119.0ms
(/ (sin b) (cos (+ a b)))

simplify787.0ms

Counts
49 → 66
Calls
49 calls:
Slowest
193.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
135.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
114.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
109.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
21.0ms
(/ (* (sin b) r) (cos (+ a b)))

prune684.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.1b

localize44.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (sin a) (sin b))
0.2b
(/ (sin b) (- (* (cos a) (cos b)) (* (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)))

rewrite38.0ms

Algorithm
rewrite-expression-head
Rules
21×add-cube-cbrt
21×*-un-lft-identity
21×add-sqr-sqrt
20×associate-*r*
18×times-frac
8×associate-/r/
8×add-exp-log
8×add-cbrt-cube
7×sin-mult
6×add-log-exp
6×cos-mult
6×pow1
3×sub-div
3×associate-/l*
3×flip--
3×frac-sub
3×associate-*l*
3×flip3--
3×associate-/r*
2×div-inv
2×*-commutative
1×cbrt-unprod
1×associate-*r/
1×prod-exp
1×pow-prod-down
1×div-exp
1×diff-log
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 89
Calls
4 calls:
Slowest
13.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
11.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
7.0ms
(* (sin a) (sin b))
6.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

series524.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
258.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
117.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
86.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
63.0ms
(* (sin a) (sin b))

simplify2.4s

Counts
70 → 101
Calls
70 calls:
Slowest
361.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
350.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
221.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
138.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (- (* (cos a) (cos b)) (* (sin a) (sin b))) (- (* (cos a) (cos b)) (* (sin a) (sin b)))) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
136.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.4s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.0b

localize15.0ms

Local error

Found 4 expressions with local error:

3.6b
(log (exp (* (sin a) (sin b))))
0.3b
(* (sin a) (sin b))
0.2b
(/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
0.2b
(* r (/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))

rewrite29.0ms

Algorithm
rewrite-expression-head
Rules
22×add-cube-cbrt
22×*-un-lft-identity
22×add-sqr-sqrt
18×associate-*r*
18×times-frac
8×add-exp-log
8×add-cbrt-cube
7×pow1
5×add-log-exp
4×associate-/r/
3×associate-/l*
3×log-pow
3×associate-*l*
3×associate-/r*
3×log-prod
2×flip--
2×div-inv
2×*-commutative
2×flip3--
1×sin-mult
1×cbrt-unprod
1×associate-*r/
1×prod-exp
1×pow-prod-down
1×rem-log-exp
1×div-exp
1×exp-to-pow
1×frac-2neg
1×clear-num
1×exp-prod
1×cbrt-undiv
Counts
4 → 85
Calls
4 calls:
Slowest
10.0ms
(/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
9.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
6.0ms
(* (sin a) (sin b))
2.0ms
(log (exp (* (sin a) (sin b))))

series460.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
232.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
123.0ms
(/ (sin b) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))))
61.0ms
(* (sin a) (sin b))
45.0ms
(log (exp (* (sin a) (sin b))))

simplify2.4s

Counts
70 → 97
Calls
70 calls:
Slowest
270.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (log (exp (* (sin a) (sin b)))) (log (exp (* (sin a) (sin b))))) (* (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
223.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (log (exp (* (sin a) (sin b)))) (log (exp (* (sin a) (sin b))))) (* (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
204.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
167.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b))))) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))) (- (* (cos a) (cos b)) (log (exp (* (sin a) (sin b)))))))
122.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))

prune1.5s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.0b

localize26.0ms

Local error

Found 4 expressions with local error:

0.3b
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
0.3b
(* (sin b) (sin a))
0.1b
(* r (sin b))
0.1b
(- (* (cos b) (cos a)) (* (sin b) (sin a)))

rewrite43.0ms

Algorithm
rewrite-expression-head
Rules
9×add-cube-cbrt
9×*-un-lft-identity
9×add-sqr-sqrt
8×add-exp-log
8×add-cbrt-cube
6×add-log-exp
6×associate-*r*
6×pow1
5×sin-mult
4×cos-mult
4×associate-/r/
3×associate-*l*
3×times-frac
3×associate-/r*
2×sub-div
2×flip--
2×frac-sub
2×*-commutative
2×flip3--
1×associate-/l*
1×div-inv
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×div-exp
1×diff-log
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 66
Calls
4 calls:
Slowest
20.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
13.0ms
(- (* (cos b) (cos a)) (* (sin b) (sin a)))
7.0ms
(* (sin b) (sin a))
2.0ms
(* r (sin b))

series470.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
285.0ms
(/ (* r (sin b)) (- (* (cos b) (cos a)) (* (sin b) (sin a))))
91.0ms
(- (* (cos b) (cos a)) (* (sin b) (sin a)))
56.0ms
(* (sin b) (sin a))
37.0ms
(* r (sin b))

simplify1.8s

Counts
42 → 78
Calls
42 calls:
Slowest
380.0ms
(- (+ (cos (+ b a)) (cos (- b a))) (- (cos (- b a)) (cos (+ b a))))
341.0ms
(- (* (+ (cos (+ b a)) (cos (- b a))) 2) (* 2 (- (cos (- b a)) (cos (+ b a)))))
196.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
165.0ms
(+ (* (* (cos b) (cos a)) (* (cos b) (cos a))) (+ (* (* (sin b) (sin a)) (* (sin b) (sin a))) (* (* (cos b) (cos a)) (* (sin b) (sin a)))))
125.0ms
(- 1 (+ (* 1/2 (pow b 2)) (* a b)))

prune941.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

regimes369.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms

end0.0ms

sample7.3s

Algorithm
intervals