Average Error: 14.7 → 0.4
Time: 22.0s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r}{\frac{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(\left(-\sin b\right) \cdot \sin a\right))_*}{\sin b}}\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.7

    \[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
  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 fma-neg0.3

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

  7. Applied associate-/l*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019010 +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: 20.2s)Debug log

sample225.0ms

Algorithm
intervals

simplify92.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
55.0ms
(/ (* r (sin b)) (cos (+ a b)))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 10.9b

localize36.0ms

Local error

Found 3 expressions with local error:

2.1b
(cos (+ a b))
0.3b
(/ (* r (sin b)) (cos (+ a b)))
0.1b
(* r (sin b))

rewrite53.0ms

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
43.0ms
(/ (* r (sin b)) (cos (+ a b)))
5.0ms
(cos (+ a b))
2.0ms
(* r (sin b))

series374.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
191.0ms
(/ (* r (sin b)) (cos (+ a b)))
146.0ms
(* r (sin b))
36.0ms
(cos (+ a b))

simplify1.1s

Counts
24 → 53
Calls
24 calls:
Slowest
571.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
204.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
142.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
84.0ms
(- (log (* r (sin b))) (log (cos (+ a b))))
13.0ms
(/ (* (sin b) r) (cos (+ a b)))

prune553.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.2b

localize29.0ms

Local error

Found 4 expressions with local error:

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

rewrite28.0ms

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×log1p-expm1-u
4×cos-mult
4×associate-/r/
4×expm1-log1p-u
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×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 → 76
Calls
4 calls:
Slowest
15.0ms
(/ (* 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))
1.0ms
(* r (sin b))

series448.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
233.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
91.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
64.0ms
(* (sin a) (sin b))
60.0ms
(* r (sin b))

simplify2.0s

Counts
45 → 88
Calls
45 calls:
Slowest
361.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
264.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
254.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
234.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
196.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune1.3s

Pruning

17 alts after pruning (17 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite10.0ms

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×associate-*r*
6×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-*l*
3×times-frac
3×associate-/r*
2×*-commutative
1×associate-/l*
1×sin-mult
1×div-inv
1×fma-udef
1×cbrt-unprod
1×prod-exp
1×pow-prod-down
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 64
Calls
4 calls:
Slowest
4.0ms
(/ (* r (sin b)) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
4.0ms
(* (sin a) (sin b))
1.0ms
(* r (sin b))
0.0ms
(fma (cos a) (cos b) (- (* (sin a) (sin b))))

series426.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
273.0ms
(/ (* r (sin b)) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
72.0ms
(fma (cos a) (cos b) (- (* (sin a) (sin b))))
46.0ms
(* (sin a) (sin b))
36.0ms
(* r (sin b))

simplify987.0ms

Counts
35 → 76
Calls
35 calls:
Slowest
278.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
171.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
81.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (fma (cos a) (cos b) (- (* (sin a) (sin b)))) (fma (cos a) (cos b) (- (* (sin a) (sin b))))) (fma (cos a) (cos b) (- (* (sin a) (sin b))))))
58.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
55.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin b) (sin a))))

prune1.1s

Pruning

16 alts after pruning (16 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (sin a) (sin b))
0.2b
(/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
0.2b
(* r (/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b))))))
0.1b
(fma (cos a) (cos b) (- (* (sin a) (sin b))))

rewrite11.0ms

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×pow1
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×associate-/l*
3×associate-*l*
3×associate-/r*
2×div-inv
2×*-commutative
1×sin-mult
1×fma-udef
1×cbrt-unprod
1×associate-*r/
1×prod-exp
1×pow-prod-down
1×div-exp
1×frac-2neg
1×clear-num
1×cbrt-undiv
Counts
4 → 83
Calls
4 calls:
Slowest
5.0ms
(* (sin a) (sin b))
3.0ms
(/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
2.0ms
(* r (/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b))))))
0.0ms
(fma (cos a) (cos b) (- (* (sin a) (sin b))))

series534.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
239.0ms
(* r (/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b))))))
141.0ms
(/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
99.0ms
(fma (cos a) (cos b) (- (* (sin a) (sin b))))
55.0ms
(* (sin a) (sin b))

simplify1.8s

Counts
60 → 95
Calls
60 calls:
Slowest
269.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
216.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (fma (cos a) (cos b) (- (* (sin a) (sin b)))) (fma (cos a) (cos b) (- (* (sin a) (sin b))))) (fma (cos a) (cos b) (- (* (sin a) (sin b))))))
201.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
143.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
75.0ms
(* (* (* (sin a) (sin a)) (sin a)) (* (* (sin b) (sin b)) (sin b)))

prune1.2s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

regimes304.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample7.5s

Algorithm
intervals