Average Error: 15.4 → 0.3
Time: 25.2s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{\cos a \cdot \cos b - \sin a \cdot \sin b}\]

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 15.4

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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

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

  9. Final simplification0.3

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

Reproduce

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

Details

Time bar (total: 23.7s)Debug log

sample544.0ms

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)

simplify16.0ms

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

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.7b

localize20.0ms

Local error

Found 3 expressions with local error:

4.5b
(cos (+ a b))
0.2b
(/ (* r (sin b)) (cos (+ a b)))
0.1b
(* r (sin b))

rewrite434.0ms

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×log1p-expm1-u
3×associate-*r*
3×times-frac
3×associate-/r*
3×pow1
3×expm1-log1p-u
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 → 44
Calls
3 calls:
Slowest
8.0ms
(/ (* r (sin b)) (cos (+ a b)))
6.0ms
(cos (+ a b))
2.0ms
(* r (sin b))

series200.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
119.0ms
(/ (* r (sin b)) (cos (+ a b)))
59.0ms
(* r (sin b))
22.0ms
(cos (+ a b))

simplify1.1s

Counts
24 → 53
Calls
24 calls:
Slowest
540.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
292.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
152.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
42.0ms
(- (log (* r (sin b))) (log (cos (+ a b))))
16.0ms
(/ (* (sin b) r) (cos (+ a b)))

prune573.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.2b

localize21.0ms

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))

rewrite25.0ms

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×log1p-expm1-u
4×associate-/r/
4×expm1-log1p-u
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×prod-diff
1×div-exp
1×diff-log
1×fma-neg
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 83
Calls
4 calls:
Slowest
10.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
7.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
4.0ms
(* (sin a) (sin b))
4.0ms
(* (cos a) (cos b))

series404.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
201.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
98.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
60.0ms
(* (cos a) (cos b))
45.0ms
(* (sin a) (sin b))

simplify2.2s

Counts
51 → 95
Calls
51 calls:
Slowest
371.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
303.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
271.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
222.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* 1/2 (pow b 2))))
220.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.3s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize7.0ms

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)))

rewrite33.0ms

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
4×log1p-expm1-u
4×expm1-log1p-u
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×prod-diff
1×div-exp
1×diff-log
1×fma-neg
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 99
Calls
4 calls:
Slowest
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)))
4.0ms
(* (sin a) (sin b))

series556.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
279.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
113.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
94.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
71.0ms
(* (sin a) (sin b))

simplify2.8s

Counts
73 → 111
Calls
73 calls:
Slowest
307.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
249.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
233.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
212.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
204.0ms
(+ (* (* (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

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite53.0ms

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×log1p-expm1-u
4×associate-/r/
4×expm1-log1p-u
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×prod-diff
1×div-exp
1×diff-log
1×fma-neg
1×frac-2neg
1×sub-neg
1×clear-num
1×cbrt-undiv
Counts
4 → 83
Calls
4 calls:
Slowest
22.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin b) (sin a))))
14.0ms
(- (* (cos a) (cos b)) (* (sin b) (sin a)))
8.0ms
(* (cos a) (cos b))
7.0ms
(* (sin b) (sin a))

series478.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
306.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin b) (sin a))))
82.0ms
(- (* (cos a) (cos b)) (* (sin b) (sin a)))
50.0ms
(* (sin b) (sin a))
39.0ms
(* (cos a) (cos b))

simplify2.1s

Counts
52 → 95
Calls
52 calls:
Slowest
353.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- b a)) (cos (+ b a)))))
290.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
212.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* 1/2 (pow b 2))))
168.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin b) (sin a)) (* (sin b) (sin a))) (* (* (cos a) (cos b)) (* (sin b) (sin a)))))
164.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- b a)) (cos (+ b a))))

prune1.4s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

regimes265.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample7.7s

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)