Average Error: 15.4 → 0.5
Time: 39.9s
Precision: 64
Internal Precision: 128
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
↓
\[\frac{r \cdot \sin b}{(\left(\sin b\right) \cdot \left(\sin a\right) + \left(\cos b \cdot \cos a\right))_*} \cdot \frac{\cos b \cdot \cos a}{\cos b \cdot \cos a - \sin a \cdot \sin b} + \left(r \cdot \frac{\sin b}{\left(\cos b \cdot \cos a\right) \cdot \left(\cos b \cdot \cos a\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)}\right) \cdot \left(\sin a \cdot \sin b\right)\]

Error

Bits error versus r

Bits error versus a

Bits error versus b

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 flip--0.4

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

  6. Applied associate-/r/0.4

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

  7. Applied associate-*r*0.4

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

  8. Simplified0.4

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

  10. Applied fma-udef0.4

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

  11. Applied distribute-rgt-in0.4

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

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

Details

Time bar (total: 38.4s)Debug log

sample248.0ms

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)

simplify22.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
21.0ms
(* r (/ (sin b) (cos (+ a b))))

prune8.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 15.7b

localize18.0ms

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))))

rewrite22.0ms

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×log1p-expm1-u
3×associate-/r*
3×pow1
3×expm1-log1p-u
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 → 63
Calls
3 calls:
Slowest
8.0ms
(/ (sin b) (cos (+ a b)))
5.0ms
(* r (/ (sin b) (cos (+ a b))))
5.0ms
(cos (+ a b))

series219.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
136.0ms
(* r (/ (sin b) (cos (+ a b))))
48.0ms
(/ (sin b) (cos (+ a b)))
34.0ms
(cos (+ a b))

simplify902.0ms

Counts
49 → 72
Calls
49 calls:
Slowest
258.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
150.0ms
(+ b (+ (* 1/3 (pow b 3)) (* a (pow b 2))))
141.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* a b)))
134.0ms
(/ (* (* (sin b) (sin b)) (sin b)) (* (* (cos (+ a b)) (cos (+ a b))) (cos (+ a b))))
34.0ms
(/ (* (cbrt (sin b)) (cbrt (sin b))) (* (cbrt (cos (+ a b))) (cbrt (cos (+ a b)))))

prune912.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.2b

localize48.0ms

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)))

rewrite65.0ms

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
23.0ms
(* 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))

series560.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
256.0ms
(* r (/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b)))))
167.0ms
(/ (sin b) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
86.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
51.0ms
(* (sin a) (sin b))

simplify3.0s

Counts
73 → 111
Calls
73 calls:
Slowest
371.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
336.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
265.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
231.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
215.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))

prune1.6s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.2b
(/ (* (sin b) r) (- (* (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))

rewrite27.0ms

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
11.0ms
(/ (* (sin b) r) (- (* (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))

series487.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
264.0ms
(/ (* (sin b) r) (- (* (cos a) (cos b)) (* (sin a) (sin b))))
114.0ms
(- (* (cos a) (cos b)) (* (sin a) (sin b)))
71.0ms
(* (sin a) (sin b))
39.0ms
(* (cos a) (cos b))

simplify2.1s

Counts
52 → 95
Calls
52 calls:
Slowest
288.0ms
(- (+ (cos (+ a b)) (cos (- a b))) (- (cos (- a b)) (cos (+ a b))))
266.0ms
(- (* (+ (cos (+ a b)) (cos (- a b))) 2) (* 2 (- (cos (- a b)) (cos (+ a b)))))
254.0ms
(+ (* (* (cos a) (cos b)) (* (cos a) (cos b))) (+ (* (* (sin a) (sin b)) (* (sin a) (sin b))) (* (* (cos a) (cos b)) (* (sin a) (sin b)))))
201.0ms
(+ (* a (* r (pow b 2))) (+ (* 1/3 (* r (pow b 3))) (* r b)))
194.0ms
(- 1 (+ (* 1/2 (pow a 2)) (* 1/2 (pow b 2))))

prune1.4s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

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

rewrite248.0ms

Algorithm
rewrite-expression-head
Rules
60×pow1
32×add-exp-log
32×add-cbrt-cube
27×pow-prod-down
22×sin-mult
22×cos-mult
17×cbrt-unprod
17×prod-exp
16×associate-/r/
14×add-cube-cbrt
14×*-un-lft-identity
14×add-sqr-sqrt
13×associate-*r/
13×associate-*l/
12×times-frac
10×frac-times
9×frac-sub
8×pow-prod-up
5×sub-div
5×associate-*r*
4×difference-of-squares
4×add-log-exp
4×log1p-expm1-u
4×pow-plus
4×associate-/r*
4×expm1-log1p-u
3×associate-/l*
3×*-commutative
3×associate-*l*
2×fma-udef
2×pow2
1×flip--
1×div-inv
1×div-exp
1×flip3--
1×distribute-lft-in
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×distribute-rgt-in
Counts
4 → 137
Calls
4 calls:
Slowest
141.0ms
(* (* r (/ (sin b) (- (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (* (sin a) (sin b)) (* (sin a) (sin b)))))) (fma (sin b) (sin a) (* (cos b) (cos a))))
56.0ms
(/ (sin b) (- (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (* (sin a) (sin b)) (* (sin a) (sin b)))))
31.0ms
(* (* (sin a) (sin b)) (* (sin a) (sin b)))
15.0ms
(* (* (cos a) (cos b)) (* (cos a) (cos b)))

series809.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
496.0ms
(* (* r (/ (sin b) (- (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (* (sin a) (sin b)) (* (sin a) (sin b)))))) (fma (sin b) (sin a) (* (cos b) (cos a))))
193.0ms
(/ (sin b) (- (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (* (sin a) (sin b)) (* (sin a) (sin b)))))
63.0ms
(* (* (sin a) (sin b)) (* (sin a) (sin b)))
57.0ms
(* (* (cos a) (cos b)) (* (cos a) (cos b)))

simplify16.1s

Counts
119 → 149
Calls
119 calls:
Slowest
671.0ms
(/ (sqrt (sin b)) (cbrt (- (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (* (sin a) (sin b)) (* (sin a) (sin b))))))
654.0ms
(/ (cbrt (sin b)) (cbrt (- (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (* (sin a) (sin b)) (* (sin a) (sin b))))))
646.0ms
(/ (cbrt (sin b)) (sqrt (- (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (* (sin a) (sin b)) (* (sin a) (sin b))))))
632.0ms
(* (* (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (cos a) (cos b))) (* (* (* (cos a) (cos a)) (cos a)) (* (* (cos b) (cos b)) (cos b))))
611.0ms
(/ (sqrt (sin b)) (sqrt (- (* (* (cos a) (cos b)) (* (cos a) (cos b))) (* (* (sin a) (sin b)) (* (sin a) (sin b))))))

prune2.3s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

regimes198.0ms

Accuracy

0% (0.4b remaining)

Error of 0.5b against oracle of 0.1b and baseline of 0.5b

bsearch6.0ms

end0.0ms

sample7.1s

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)