Average Error: 14.9 → 0.3
Time: 20.8s
Precision: 64
Internal Precision: 128
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{\sin b}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-(e^{\log_* (1 + \sin a \cdot \sin b)} - 1)^*\right))_*} \cdot r\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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 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 *-un-lft-identity0.3

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  11. Applied expm1-log1p-u0.3

    \[\leadsto r \cdot \frac{\sin b}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-\color{blue}{(e^{\log_* (1 + \sin a \cdot \sin b)} - 1)^*}\right))_*}\]

  12. Final simplification0.3

    \[\leadsto \frac{\sin b}{(\left(\cos a\right) \cdot \left(\cos b\right) + \left(-(e^{\log_* (1 + \sin a \cdot \sin b)} - 1)^*\right))_*} \cdot r\]

Reproduce

herbie shell --seed 2019022 +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: 18.9s)Debug log

sample265.0ms

Algorithm
intervals
Results
92.0ms95×body1280valid
69.0ms52×body640valid
14.0ms28×body320valid
13.0ms73×body80valid
2.0ms8×body160valid

simplify50.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
49.0ms
(/ (* r (sin b)) (cos (+ a b)))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 13.0b

localize18.0ms

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))

rewrite15.0ms

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
5.0ms
(/ (* r (sin b)) (cos (+ a b)))
4.0ms
(cos (+ a b))
1.0ms
(* r (sin b))

series149.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
91.0ms
(/ (* r (sin b)) (cos (+ a b)))
37.0ms
(* r (sin b))
20.0ms
(cos (+ a b))

simplify1.0s

Counts
24 → 53
Calls
24 calls:
Slowest
430.0ms
(/ (* (* (* r (sin b)) (* r (sin b))) (* r (sin b))) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
307.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
162.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
23.0ms
(/ (* (sin b) r) (cos (+ a b)))
22.0ms
(- (log (* r (sin b))) (log (cos (+ a b))))

prune576.0ms

Pruning

6 alts after pruning (6 fresh and 0 done)

Merged error: 0.2b

localize30.0ms

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)))

rewrite47.0ms

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
24.0ms
(/ (* r (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))
2.0ms
(* r (sin b))

series460.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
228.0ms
(/ (* r (sin b)) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
118.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
58.0ms
(* (sin a) (sin b))
57.0ms
(* r (sin b))

simplify1.9s

Counts
45 → 88
Calls
45 calls:
Slowest
339.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
294.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
293.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
218.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
121.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))

prune1.1s

Pruning

13 alts after pruning (13 fresh and 0 done)

Merged error: 0.0b

localize9.0ms

Local error

Found 4 expressions with local error:

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

rewrite11.0ms

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
6.0ms
(/ (* r (sin b)) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
3.0ms
(* (sin a) (sin b))
1.0ms
(* r (sin b))
0.0ms
(fma (cos a) (cos b) (- (* (sin a) (sin b))))

series457.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
237.0ms
(/ (* r (sin b)) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
110.0ms
(fma (cos a) (cos b) (- (* (sin a) (sin b))))
57.0ms
(* r (sin b))
53.0ms
(* (sin a) (sin b))

simplify921.0ms

Counts
35 → 76
Calls
35 calls:
Slowest
294.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
160.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
69.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
63.0ms
(* (* (* (sin a) (sin a)) (sin a)) (* (* (sin b) (sin b)) (sin b)))
55.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin b) (sin a))))

prune1.1s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

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

rewrite21.0ms

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
9.0ms
(* (sin a) (sin b))
7.0ms
(/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
3.0ms
(* r (/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b))))))
0.0ms
(fma (cos a) (cos b) (- (* (sin a) (sin b))))

series527.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
272.0ms
(* r (/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b))))))
125.0ms
(/ (sin b) (fma (cos a) (cos b) (- (* (sin a) (sin b)))))
74.0ms
(fma (cos a) (cos b) (- (* (sin a) (sin b))))
55.0ms
(* (sin a) (sin b))

simplify1.6s

Counts
60 → 95
Calls
60 calls:
Slowest
245.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
179.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
163.0ms
(/ (* (* (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))))))
141.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
64.0ms
(* (* (* (sin a) (sin a)) (sin a)) (* (* (sin b) (sin b)) (sin b)))

prune1.3s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0b

regimes365.0ms

Accuracy

0% (0.3b remaining)

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

bsearch3.0ms

end0.0ms

sample7.0s

Algorithm
intervals
Results
3.4s2911×body1280valid
2.0s1856×body640valid
761.0ms2119×body80valid
376.0ms763×body320valid
182.0ms353×body160valid