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(12c Platinum) Fast & Accurate Trigonometric Functions
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(12c Platinum) Fast & Accurate Trigonometric Functions
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(12c Platinum) Fast & Accurate Trigonometric Functions
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12-10-2021, 05:12 PM
Post: #1
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(12c Platinum) Fast & Accurate Trigonometric Functions
RPN Program:
001- 2 023- CHS 045- g x≤y 067- / 089- g GTO 265 002- STO 6 024- g x≤y 046- g GTO 017 068- + 090- 2 003- R↓ 025- g GTO 031 047- R↓ 069- RCL 8 091- STO 6 004- RCL .0 026- 2 048- x⇄y 070- × 092- R↓ 005- ÷ 027- × 049- STO 8 071- ENTER 093- RCL .0 006- x⇄y 028- x⇄y 050- g x² 072- g x² 094- ÷ 007- STO 4 029- - 051- ENTER 073- 4 095- CHS 008- x⇄y 030- ENTER 052- ENTER 074- × 096- 9 009- ENTER 031- R↓ 053- ENTER 075- CHS 097- 0 010- g x² 032- ENTER 054- RCL .2 076- 3 098- + 011- g √ᵡ 033- g x² 055- × 077- + 099- g GTO 006 012- 3 034- g √ᵡ 056- RCL .3 078- × 100- RCL .0 013- 6 035- ENTER 057- - 079- RCL 6 101- ÷ 014- 0 036- ENTER 058- × 080- 2 102- STO 4 015- g x≤y 037- 9 059- RCL .4 081- g x≤y 103- x⇄y 016- g GTO 120 038- 0 060- + 082- g GTO 087 104- STO 7 017- R↓ 039- - 061- × 083- R↓ 105- 1 018- R↓ 040- g x=0 062- RCL .5 084- g x=0 106- STO 6 019- 9 041- g GTO 047 063- - 085- g GTO 114 107- RCL 4 020- 0 042- R↓ 064- × 086- g GTO 109 108- g GTO 009 021- g x≤y 043- 9 065- RCL .1 087- R↓ 109- CLx 022- g GTO 026 044- 0 066- 3 088- x⇄y 110- STO 6 111- x⇄y 152- 0 193- g GTO 196 234- + 275- g GTO 000 112- RCL 4 153- × 194- 1/x 235- × 113- g GTO 095 154- RCL .0 195- 9 236- RCL .8 114- x⇄y 155- × 196- 0 237- - 115- RCL 4 156- g GTO 265 197- RCL .1 238- × optional steps: 116- x⇄y 157- x⇄y 198- × 239- RCL .9 117- ÷ 158- STO 4 199- STO 6 240- + 276- 1.745329252 118- RCL 7 159- x⇄y 200- x⇄y 241- × 287- EEX 119- g GTO 266 160- 2 201- 2 242- 3 288- 2 120- x⇄y 161- STO 7 202- ENTER 243- 1/x 289- CHS 121- EEX 162- R↓ 203- 3 244- - 290- STO .1 122- 1 163- 1 204- g √ᵡ 245- × 291- 2.1 123- 2 164- x⇄y 205- STO 9 246- 1 294- EEX 124- g x≤y 165- + 206- - 247- + 295- 26 125- g GTO 146 166- g x=0 207- x⇄y 248- RCL 9 297- CHS 126- R↓ 167- g GTO 268 208- g x≤y 249- × 298- STO .2 127- x⇄y 168- 1 209- g GTO 221 250- RCL 8 299- 4.47569 128- ÷ 169- g LSTx 210- ENTER 251- + 306- EEX 129- g LSTx 170- - 211- ENTER 252- RCL 6 307- 20 130- x⇄y 171- 1 212- RCL 9 253- g x=0 309- CHS 131- g INTG 172- g LSTx 213- × 254- x⇄y 310- STO .3 132- × 173- + 214- 1 255- x⇄y 311- 5.55391606 133- g x≤y 174- ÷ 215- - 256- - 321- EEX 134- CHS 175- g √ᵡ 216- x⇄y 257- RCL 5 322- 14 135- + 176- g GTO 183 217- RCL 9 258- × 324- CHS 136- g GTO 019 177- x⇄y 218- + 259- RCL 7 325- STO .4 137- x⇄y 178- STO 4 219- ÷ 260- × 326- 3.281837614 138- STO 4 179- x⇄y 220- 3 261- RCL .1 337- EEX 139- x⇄y 180- 1 221- 0 262- ÷ 338- 8 140- ENTER 181- STO 7 222- RCL .1 263- RCL .0 339- CHS 141- g x² 182- x⇄y 223- × 264- × 340- STO .5 142- 1 183- ENTER 224- STO 8 265- RCL 4 341- .0784 143- - 184- g x² 225- x⇄y 266- x⇄y 346- CHS 144- g x=0 185- g √ᵡ 226- STO 9 267- g GTO 000 347- STO .6 145- g GTO 150 186- g x=0 227- g x² 268- RCL 7 348- .110351 146- CHS 187- g GTO 265 228- ENTER 269- g GTO 151 355- STO .7 147- g √ᵡ 188- ÷ 229- ENTER 270- 1 356- .142838 148- ÷ 189- STO 5 230- ENTER 271- g GTO 273 363- STO .8 149- g GTO 180 190- 1 231- RCL .6 272- RCL .1 364- .199999839 150- x⇄y 191- g LSTx 232- × 273- STO .0 374- STO .9 151- 9 192- g x≤y 233- RCL .7 274- R↓ 375- g GTO 270 Usage: g GTO 276 R/S if the optional steps have been keyed in this will load all the constants needed by the program into registers R.0 through R.9, otherwise store them manually: R.0: 1 (DEG) R.1: 1.745329252E-02 (pi/180) R.2: 2.1E-26 R.3: 4.47569E-20 R.4: 5.55391606E-14 R.5: 3.281837614E-08 (sine polynomial coefficients) R.6: -0.0784 R.7: 0.110351 R.8: 0.142838 R.9: 0.199999839 (arctangent polynomial coefficients) SIN: R/S COS: g GTO 090 R/S TAN: g GTO 100 R/S ASIN: g GTO 137 R/S ACOS: g GTO 157 R/S ATAN: g GTO 177 R/S DEG: g GTO 270 R/S RAD: g GTO 272 R/S GRAD: .9 1/x STO .0 Input ranges: SIN, COS, TAN: |x| < 1E12 (degrees) ASIN, ACOS: |x| ≤ 1 ATAN: -1E-50 < x < 1E50 Examples: First set the display to FIX 9 and DEG mode, if not already set: f 9 g GTO 270 R/S 1) asin(acos(atan(tan(cos(sin(9)))))) Keystrokes Display 9 9. R/S 0.156434465 g GTO 090 R/S 0.999996273 g GTO 100 R/S 0.017455000 g GTO 177 R/S 0.999996273 g GTO 157 R/S 0.156434568 g GTO 137 R/S 9.000005947 This is a standard test created by Mike Sebastian to check trigonometric algorithms in calculators (Calculator Forensics). 2) sin²(30) + cos²(30): Notice that the stack register X is always preserved, so there's no need to store intermediate results in simple chain calculations (one-level only), as in this example. Keystrokes Display 30 R/S 0.500000000 g x² 0.250000000 30 g GTO 090 R/S 0.866025404 g x² 0.750000000 + 1.000000000 3) sin(-1234567.8) + acos(-0.77) Keystrokes Display 1234567.8 CHS -1,234,567.8 R/S -0.790155012 .77 CHS g GTO 157 R/S 140.3538889 + 139.5637338 4) tan(-6.2 rad)/asin(0.15 rad) Set the angular mode to RADIANS: g GTO 272 R/S Keystrokes Display 6.2 CHS g GTO 100 R/S 0.083377715 0.15 g GTO 137 R/S 0.150568273 ÷ 0.553753545 Go back to DEG mode: g GTO 270 R/S 0.553753545 Notice that the first example results do not quite match the ones you would get on the classic scientific Voyagers. As a matter of fact, the final result is even closer to 9: 9.00000594679 (which is rounded to 10 digits to fit in the display). Unlike the classic Voyagers, the full 12-digit mantissa is not rounded to the number of digits in the display. Since the HP 12c Platinum has abandoned the old HP rounding philosophy, it does not make much sense to implement this feature, but we can use the remaining programming space to replicate the original HP rounding method, just for testing purposes: 1) Change the line 267 to g GTO 376: g GTO 266 f P/R g GTO 376 f P/R 2) Key in these lines: 376- ENTER 377- g x² 378- g √ᵡ 379- g x=0 380- g GTO 000 381- g LN 382- 1 383- 0 384- g LN 385- ÷ 386- g INTG 387- 1 388- - 389- 1 390- 0 391- x⇄y 392- yᵡ 393- ÷ 394- g LSTx 395- x⇄y 396- f RND 397- x⇄y 398- × 399- g GTO 000 This will round the answers to the number of the decimal places of the display plus one. Let us repeat the first example: 1) asin(acos(atan(tan(cos(sin(9)))))), assuming FIX 9 and DEG mode Keystrokes Display 9 9. R/S 0.156434465 g GTO 090 R/S 0.999996273 g GTO 100 R/S 0.017455000 g GTO 177 R/S 0.999996272 g GTO 157 R/S 0.156441660 g GTO 137 R/S 9.000417403 Now all 10-digit results exactly match those of the HP-15C and other HP 10-digit calculators. Don't forget to revert to the original program by editing the line 267: g GTO 266 f P/R g GTO 000 f P/R Formulae: \(\sin x =r\left ( 3-4r^{2} \right )\) where \(r = x\left (a_{1}+x^{2}\left (a_{2}+x^{2}\left (a_{3}+x^{2}\left (a_{4}+a_{5}x^{2}\right )\right )\right) \right )\) and where a₁ = 5.81776417333E-03 a₂ =-3.281837614E-08 a₃ = 5.55391606E-14 a₄ =-4.47569E-20 a₅ = 2.1E-26 x in degrees; |x| ≤ 90 degrees; Max Abs Error < 1.46E-12 \(\arctan x =x\left ( b_{1} + x^{2}\left ( b_{2}+x^{2}\left (b_{3}+x^{2}\left ( b_{4}+x^{2}\left ( b_{5}+b_{6} x^{2}\right ) \right )\right ) \right )\right )\) where b₁ = 1 b₂ =-0.333333333333 b₃ = 0.199999839 b₄ =-0.142838 b₅ = 0.110351 b₆ =-0.0784 |x| ≤ \(2-\sqrt{3}\); Max Abs Error < 7.85E-12; results in radians \(\arctan x =\frac{x}{|x|}\left ( \frac{\pi }{6}+\arctan \frac{|x|\sqrt{3}-1}{|x|+\sqrt{3}} \right )\) \(2-\sqrt{3}\) < |x| ≤ 1 \(\arctan x =\frac{x}{|x|}\left ( \frac{\pi }{2}-\arctan \frac{1}{|x|} \right )\) |x| > 1 \(\cos x =\sin \left ( \frac{\pi }{2} - x \right )\) \(\tan x=\frac{\sin x}{\cos x}\) \(\arcsin x = |x|\arctan \frac{x}{\sqrt{1-x^{2}}}\) |x| < 1 \(\arccos x =2\arctan \sqrt{\frac{1-x}{1+x}}\) -1 < x ≤ 1 |
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(12c Platinum) Fast & Accurate Trigonometric Functions - Gerson W. Barbosa - 12-10-2021 05:12 PM
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