Sin(1°) (1 Mrd. Stellen; accurate to 1 billion digits)

  

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Sin(1°) = 0.01745240643728351281941897851631619247225272030713964268361242764...

a) Calc Sin(1°) with Mathematica 20874452 digits:


§0: N[Sin[1 Degree], 20874452];
§1: N[Root[{1 - 384 #^2 + 7040 #^4 - 50176 #^6 + 182784 #^8 - 372736 #^10 + 430080 #^12 - 262144 #^14 + 65536 #^16 &, # -3 #2 + 4 #2^3 &}, {9, 2}],20874452]
§2: N[Root[1-24#1-144#1^2+248#1^3+1680#1^4-864#1^5-7168#1^6+1152#1^7+13824#1^8-512#1^9-12288#1^10+4096#1^12&,7],20874452];Sin(1°)=Sqrt[(1-Sqrt[1-%^2])*2^(-1)];
§3: w2=N[Sqrt[2],20874452];w3=N[Sqrt[3],20874452];w5=N[Sqrt[5],20874452];w35=(1-w3)(Sqrt[5+w5]/8)+w2(w5-1)((w3+1)/16);Sin(1°)=Root[w35-3#+4#^3&,2]
§4: Householder Iteration (quart. convergence http://numbers.computation.free.fr/Constants/Algorithms/newton.html )

b) Calc Sin(1°) with NumberFactory (YMP c++ -> new code)



Benchmark 20874452 digits:
Software   |  Algor.   |  Time
-----------+-----------+---------
Mathematica| §0        |126.14 s
Mathematica| §1        | 51.28 s
Mathematica| §2        | 27.44 s
Mathematica| §3        | 18.67 s
Mathematica| §4        | 16.06 s
YMP + Gerd | §4        |  1.60 s



Benchmark 333 Mio. digits:
Software   |  Algor.   |  Time
-----------+-----------+---------
Mathematica| §1        |1097 s
Mathematica| §4        | 372 s
YMP + Gerd | §4        |  24 s (RAM-Disk)





Benchmark 10^9 digits (ger.: 1 Mrd.; engl. 1 bio.):
Software   |  Algor.   |  Time
-----------+-----------+---------
YMP + Gerd | §4        |  85 s (RAM-Disk)












Gerd Lamprecht, Deutschland 03.12.2024






Date Announced Date Completed: Source: Who: Constant: Decimal Digits: Time: Computer:
November 26, 2024 November 26, 2024 Source Gerd Lamprecht sin(10°) 1,000,000,000 Compute:  28.6 min
Verify: 1.2 min		 Intel Core i9 7900X
32 GB
November 24, 2024 November 24, 2024 Source Gerd Lamprecht sin(3°) 3,000,000,000 Compute:  37 s
Verify: 28 s		 Intel Core i9 7900X
32 GB
November 13, 2024 November 13, 2024 Source Gerd Lamprecht Gamma(1/10) 3,000,000,000 Compute:  46 minutes
Not Verified		 Intel Core i9 7900X
32 GB
November 10, 2024 November 10, 2024 Source Gerd Lamprecht Gamma(1/24) 3,000,000,000 Compute:  17 minutes
Not Verified		 Intel Core i9 7900X
32 GB
November 07, 2024 November 07, 2024 Source Gerd Lamprecht Gamma(1/8) 4,000,000,000 Compute:  28 minutes
Verify: 17 min		 Intel Core i9 7900X
32 GB
October 20, 2024 October 19, 2024 Source Gerd Lamprecht Gamma(1/5) 3,000,000,010 Compute:  48.5 minutes
Verified by Alexander Yee		 Intel Core i9 7900X
32 GB

Links:
03.11.2024: ExotischeGammawerte-Gamma1d5 (3 Mrd. Stellen {3 bio digits})
07.11.2024: New-formula-Gamma1d8 (4 Mrd. Stellen {4 bio digits})
10.11.2024: https: Liste-mathematische-Rekorde
10.11.2024: http: Liste-mathematische-Rekorde
10.11.2024: y-cruncher/records