Average Error: 15.4 → 0.4
Time: 26.2s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{\cos b \cdot \cos a - \sin b \cdot \sin a}{\sin b}}\]

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 15.4

    \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
  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 associate-*r/0.3

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

  7. Applied associate-/l*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019016 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))

Details

Time bar (total: 24.6s)Debug log

sample281.0ms

Algorithm
intervals
Results
256×(pre true 80)
98×(body real 1280)
71×(body real 80)
51×(body real 640)
20×(body real 320)
16×(body real 160)

simplify27.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
27.0ms
(* r (/ (sin b) (cos (+ a b))))

prune13.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.7b

localize60.0ms

Local error

Found 3 expressions with local error:

4.5b
(cos (+ a b))
0.2b
(/ (sin b) (cos (+ a b)))
0.2b
(* r (/ (sin b) (cos (+ a b))))

rewrite21.0ms

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

series164.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
98.0ms
(* r (/ (sin b) (cos (+ a b))))
35.0ms
(/ (sin b) (cos (+ a b)))
30.0ms
(cos (+ a b))

simplify773.0ms

Counts
49 → 66
Calls
49 calls:
Slowest
195.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
142.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
130.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
91.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
18.0ms
(/ (* (cbrt (sin b)) (cbrt (sin b))) (* (cbrt (cos (+ a b))) (cbrt (cos (+ a b)))))

prune705.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.2b

localize35.0ms

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)))

rewrite64.0ms

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
22.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
20.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
13.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
7.0ms
(* (sin a) (sin b))

series637.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
295.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
144.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
120.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
78.0ms
(* (sin a) (sin b))

simplify2.6s

Counts
70 → 101
Calls
70 calls:
Slowest
391.0ms
(/ (sin b) (cbrt (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
386.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
290.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
174.0ms
(/ (* (* (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)))))
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)))))

prune1.5s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

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

rewrite47.0ms

Algorithm
rewrite-expression-head
Rules
10×add-cube-cbrt
10×add-exp-log
10×add-cbrt-cube
10×*-un-lft-identity
10×add-sqr-sqrt
8×pow1
6×add-log-exp
6×associate-*l*
6×associate-*r*
5×sin-mult
5×cos-mult
4×associate-/r/
3×times-frac
3×associate-/r*
2×sub-div
2×flip--
2×cbrt-unprod
2×frac-sub
2×*-commutative
2×prod-exp
2×pow-prod-down
2×flip3--
1×associate-/l*
1×div-inv
1×div-exp
1×diff-log
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 73
Calls
4 calls:
Slowest
19.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
13.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
10.0ms
(* (sin a) (sin b))
4.0ms
(* (cos a) (cos b))

series460.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
230.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
97.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
74.0ms
(* (sin a) (sin b))
59.0ms
(* (cos a) (cos b))

simplify1.8s

Counts
48 → 85
Calls
48 calls:
Slowest
352.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
304.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
206.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
148.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* 1/2 (pow b 2))))
116.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune1.1s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize6.0ms

Local error

Found 4 expressions with local error:

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

rewrite42.0ms

Algorithm
rewrite-expression-head
Rules
21×add-cube-cbrt
21×*-un-lft-identity
21×add-sqr-sqrt
18×times-frac
16×associate-/r*
8×add-exp-log
8×add-cbrt-cube
6×add-log-exp
6×pow1
5×sin-mult
4×associate-/l/
4×cos-mult
3×associate-/l*
3×div-inv
3×associate-*l*
3×associate-*r*
2×sub-div
2×flip--
2×frac-sub
2×flip3--
2×frac-2neg
2×clear-num
1×cbrt-unprod
1×*-commutative
1×associate-/r/
1×prod-exp
1×pow-prod-down
1×div-exp
1×div-sub
1×diff-log
1×sub-neg
1×cbrt-undiv
Counts
4 → 88
Calls
4 calls:
Slowest
26.0ms
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
9.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
3.0ms
(* (sin a) (sin b))
2.0ms
(/ r (/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b)))

series554.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
279.0ms
(/ r (/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b)))
112.0ms
(/ (- (* (cos a) (cos b)) (* (sin a) (sin b))) (sin b))
95.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
67.0ms
(* (sin a) (sin b))

simplify4.1s

Counts
69 → 100
Calls
69 calls:
Slowest
356.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
330.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
324.0ms
(- (pow (* (cos a) (cos b)) 3) (pow (* (sin a) (sin b)) 3))
319.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
308.0ms
(+ (* 1/2 (* (pow a 2) (* r b))) (+ (* a (* r (pow b 2))) (* r b)))

prune1.5s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

regimes298.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample7.9s

Algorithm
intervals
Results
8000×(pre true 80)
2933×(body real 1280)
2098×(body real 80)
1786×(body real 640)
813×(body real 320)
370×(body real 160)