Details

simplify11.1m (27.3%)

prune10.8m (26.6%)

sample9.5m (23.4%)

Algorithm
90×intervals
Results
6.5m100771×body10240exit
1.1m398158×body80valid
38.1s26187×body1280valid
19.0s20296×body640valid
11.6s70024×body80nan
6.9s10678×body320valid
5.5s1670×body1280nan
4.7s9516×body160valid
3.7s1380×body640nan
2.6s49536×pre80true
1.6s1426×body2560valid
1.4s662×body320nan
818.0ms337×body160nan
567.0ms628×body5120valid

rewrite7.4m (18.2%)

Algorithm
146×rewrite-expression-head
Calls

454 calls:

Slowest
1.2m
(* (sqrt (* 2 PI)) (* (* (/ (pow (+ (- 7 z) 0.5) (+ 0.5 (- 0 z))) (exp (+ (- 7 z) 0.5))) (+ (+ (+ (/ 1.5056327351493116e-07 (- 8 z)) (/ 9.984369578019572e-06 (- 7 z))) (+ (+ (/ -0.13857109526572012 (- 6 z)) 0.9999999999998099) (+ (/ 676.5203681218851 (- 1 z)) (+ (/ -1259.1392167224028 (- 2 z)) (/ 771.3234287776531 (- 3 z)))))) (+ (/ -176.6150291621406 (- 4 z)) (/ 12.507343278686905 (- 5 z))))) (/ PI (sin (* PI z)))))
24.3s
(* (/ (* (pow (- (+ z 0.5) -6) (+ (- z 1) 0.5)) (sqrt (* PI 2))) (exp (- (+ z 0.5) -6))) (+ (+ (+ (/ -0.13857109526572012 (- (+ 6 z) 1)) (/ 12.507343278686905 (+ z 4))) (+ (/ 9.984369578019572e-06 (+ 6 z)) (/ 1.5056327351493116e-07 (+ 7 z)))) (+ (+ (/ 676.5203681218851 z) (+ (/ -176.6150291621406 (- z -3)) (+ (/ 771.3234287776531 (+ 2 z)) (/ -1259.1392167224028 (+ z 1))))) 0.9999999999998099)))
24.3s
(* (/ (/ (* (pow (- (+ z 0.5) -6) (+ (- z 1) 0.5)) (sqrt (* PI 2))) (exp (+ z 0.5))) (exp (- -6))) (+ (+ (+ (/ -0.13857109526572012 (- (+ 6 z) 1)) (/ 12.507343278686905 (+ z 4))) (+ (/ 9.984369578019572e-06 (+ 6 z)) (/ 1.5056327351493116e-07 (+ 7 z)))) (+ (+ (/ 676.5203681218851 z) (+ (/ -176.6150291621406 (- z -3)) (+ (/ 771.3234287776531 (+ 2 z)) (/ -1259.1392167224028 (+ z 1))))) 0.9999999999998099)))
24.2s
(* (/ (* (pow (- (+ z 0.5) -6) (+ (- z 1) 0.5)) (sqrt (* PI 2))) (exp (+ z 0.5))) (+ (+ (+ (/ -0.13857109526572012 (- (+ 6 z) 1)) (/ 12.507343278686905 (+ z 4))) (+ (/ 9.984369578019572e-06 (+ 6 z)) (/ 1.5056327351493116e-07 (+ 7 z)))) (+ (+ (/ 676.5203681218851 z) (+ (/ -176.6150291621406 (- z -3)) (+ (/ 771.3234287776531 (+ 2 z)) (/ -1259.1392167224028 (+ z 1))))) 0.9999999999998099)))
16.4s
(* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x)
Rules
3743181×frac-times
3263046×frac-add
2078268×flip-+ flip3-+
1849278×frac-sub
1445131×flip3-- flip--
1383514×associate-*l/
1314724×associate-*r/
52977×*-un-lft-identity
42144×times-frac
21421×add-sqr-sqrt
20163×distribute-lft-out
17272×add-cube-cbrt
14301×sqrt-prod
4987×prod-diff
3398×add-exp-log
3056×associate-*r*
3027×pow1
2769×distribute-rgt-in distribute-lft-in
2406×add-cbrt-cube
1670×associate-/r/
1573×difference-of-squares
1517×associate-*l*
1502×prod-exp
1460×distribute-lft-out--
1364×associate-/l*
1029×cbrt-unprod
1011×div-inv
890×pow-prod-down
824×div-exp
614×add-log-exp
611×sqrt-div
610×pow-prod-up
592×associate-/r*
533×fma-def
455×expm1-log1p-u
454×insert-posit16 log1p-expm1-u
431×cbrt-undiv
425×cbrt-div
363×exp-sum
321×unswap-sqr
279×log-prod
268×cbrt-prod
249×pow-plus
202×swap-sqr
200×unpow-prod-down
193×pow-sqr
184×pow-div
174×fma-neg
171×*-commutative
170×log-pow
148×pow2
145×pow-exp
133×associate-/l/
117×1-exp
105×rec-exp
104×pow-flip
102×frac-2neg clear-num
100×inv-pow
92×sqrt-pow1
91×sub-neg
73×pow1/2
69×associate--r+
67×sqr-pow pow-to-exp
64×pow1/3
55×fma-udef
46×sum-log exp-prod
44×diff-log
25×pow3
22×cube-unmult div-sub
19×rem-sqrt-square
18×+-commutative
16×un-div-inv distribute-rgt-out
13×distribute-rgt1-in hypot-udef
11×sin-sum
10×rem-log-exp
associate--l+ hypot-def
unpow-prod-up exp-diff associate-+l- sin-mult
associate-+l+ sqrt-unprod pow-sub
pow-unpow
log-div
pow-pow associate--r-
associate-+r+ distribute-lft-neg-in distribute-rgt-neg-in rem-exp-log rem-square-sqrt
neg-sub0 sub0-neg sqr-sin exp-to-pow distribute-rgt-out-- exp-neg distribute-neg-in rem-cbrt-cube
expm1-log1p expm1-udef associate--l- associate-+r- distribute-rgt-neg-out log1p-expm1 neg-mul-1 cos-sum log1p-udef unsub-neg

series1.6m (3.8%)

Calls

454 calls:

Slowest
3.3s
(* (* (/ (/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) (sqrt 1.0))) (/ (sqrt (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))) 1)) (/ (sqrt (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i)))) (- (+ (+ alpha beta) (* 2 i)) (sqrt 1.0))))
3.2s
(* (/ (/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) (sqrt 1.0))) (/ (/ (sqrt (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (/ (+ (+ alpha beta) (* 2 i)) (sqrt (+ (* beta alpha) (* i (+ (+ alpha beta) i)))))) (- (+ (+ alpha beta) (* 2 i)) (sqrt 1.0))))
3.2s
(* (/ (/ (* i (+ (+ alpha beta) i)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) (sqrt 1.0))) (/ (/ (+ (* beta alpha) (* i (+ (+ alpha beta) i))) (+ (+ alpha beta) (* 2 i))) (- (+ (+ alpha beta) (* 2 i)) (sqrt 1.0))))
2.0s
(* (cbrt (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) -0.284496736)) (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429)))))))) (exp (- (* (fabs x) (fabs x))))))) (cbrt (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) -0.284496736)) (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429)))))))) (exp (- (* (fabs x) (fabs x))))))))
1.4s
(* (pow (* -2 (log u1)) 0.5) 1/6)

localize7.2s (0.3%)

regimes7.2s (0.3%)

Accuracy

Total 32.4b remaining (27.5%)

Threshold costs 4.3b (3.6%)

6.9b33.2%_divideComplex, imaginary part
6.9b36.1%_divideComplex, real part
5.3b15.8%Octave 3.8, jcobi/2
3.2b74.4%Octave 3.8, jcobi/1
1.7b0%Jmat.Real.lambertw, newton loop step

bsearch2.0s (0.1%)

end2.0ms (0.0%)