Average Error: 14.9 → 0.3
Time: 21.8s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

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 14.9

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

  3. Applied cos-sum0.3

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

  5. Applied *-un-lft-identity0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}\]

  6. Applied times-frac0.3

    \[\leadsto \color{blue}{\frac{r}{1} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]

  7. Simplified0.3

    \[\leadsto \color{blue}{r} \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

  8. Taylor expanded around -inf 0.3

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{\sin a \cdot \sin b}}\]

  9. Final simplification0.3

    \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

Reproduce

herbie shell --seed 2019022 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))

Details

Time bar (total: 20.0s)Debug log

sample332.0ms

Algorithm
intervals
Results
109.0ms95×body1280valid
44.0ms52×body640valid
36.0ms8×body160valid
16.0ms28×body320valid
15.0ms73×body80valid

simplify17.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
16.0ms
(/ (* r (sin b)) (cos (+ a b)))

prune10.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.0b

localize24.0ms

Local error

Found 3 expressions with local error:

2.8b
(cos (+ a b))
0.2b
(/ (* r (sin b)) (cos (+ a b)))
0.1b
(* r (sin b))

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×add-sqr-sqrt
5×add-exp-log
5×add-cbrt-cube
3×add-log-exp
3×associate-*r*
3×times-frac
3×associate-/r*
3×pow1
1×associate-/l*
1×div-inv
1×*-commutative
1×div-exp
1×frac-2neg
1×clear-num
1×cos-sum
1×cbrt-undiv
Counts
3 → 38
Calls
3 calls:
Slowest
6.0ms
(/ (* r (sin b)) (cos (+ a b)))
3.0ms
(cos (+ a b))
2.0ms
(* r (sin b))

series199.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
109.0ms
(/ (* r (sin b)) (cos (+ a b)))
56.0ms
(* r (sin b))
33.0ms
(cos (+ a b))

simplify951.0ms

Counts
24 → 47
Calls
24 calls:
Slowest
555.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
211.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
85.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
40.0ms
(- (log (* r (sin b))) (log (cos (+ a b))))
11.0ms
(/ (* (sin b) r) (cos (+ a b)))

prune523.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.2b

localize40.0ms

Local error

Found 4 expressions with local error:

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

rewrite38.0ms

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
19.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
9.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
7.0ms
(* (sin a) (sin b))
2.0ms
(* r (sin b))

series423.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
234.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
91.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
53.0ms
(* (sin a) (sin b))
46.0ms
(* r (sin b))

simplify1.5s

Counts
42 → 78
Calls
42 calls:
Slowest
306.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
250.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
172.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
134.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
110.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune994.0ms

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.0b

localize1.0ms

Local error

Found 4 expressions with local error:

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

rewrite22.0ms

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
10.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
6.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
3.0ms
(* (sin a) (sin b))
1.0ms
(* r (sin b))

series414.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
215.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
81.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
72.0ms
(* (sin a) (sin b))
46.0ms
(* r (sin b))

simplify1.6s

Counts
42 → 78
Calls
42 calls:
Slowest
380.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
317.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
170.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
124.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
95.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.0s

Pruning

13 alts after pruning (12 fresh and 1 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

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
(* (sin a) (sin b))
0.1b
(- (* (cos a) (cos b)) (* (sin a) (sin b)))

rewrite63.0ms

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
21.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
19.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
14.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
7.0ms
(* (sin a) (sin b))

series546.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
268.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
137.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
88.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
52.0ms
(* (sin a) (sin b))

simplify2.2s

Counts
70 → 101
Calls
70 calls:
Slowest
348.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
305.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
170.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
153.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
141.0ms
(/ (* (* (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)))))

prune1.3s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

regimes283.0ms

Accuracy

0% (0.3b remaining)

Error of 0.3b against oracle of 0.1b and baseline of 0.3b

bsearch4.0ms

end0.0ms

sample7.5s

Algorithm
intervals
Results
3.6s2911×body1280valid
2.5s1856×body640valid
505.0ms763×body320valid
490.0ms2119×body80valid
177.0ms353×body160valid