## Monday, October 27, 2014

### More Fun with HP 12C (sine, cosine, error function approximations)

Fun with HP 12C Part II

Sine Approximation

Source:  Ted’s Math World, Calculator Magic #4

Retrieved 2014-10-13

sin x ≈ (((x^2/72 – 1)*x^2/42 + 1)*x^/20 – 1)*x^2/6 + 1)*x

Registers:  R0 = x in radians, R1 = sin x

 STEP KEY CODE STEP KEY CODE 01 STO 0 44, 0 21 0 0 02 2 2 22 ÷ 10 03 y^x 21 23 * 20 04 ENTER 36 24 1 1 05 ENTER 36 25 - 30 06 ENTER 36 26 X<>Y 34 07 7 7 27 6 6 08 2 2 28 ÷ 10 09 ÷ 10 29 * 20 10 1 1 30 1 1 11 - 30 31 + 40 12 X<>Y 34 32 RCL 0 45, 0 13 4 4 33 * 20 14 2 2 34 STO 1 44, 1 15 ÷ 10 35 GTO 00 43, 33, 00 16 * 20 17 1 1 18 + 40 19 X<>Y 34 20 2 2

Examples:

sin 0.25 ≈ 0.2474

sin -1.82 ≈ -0.9691

Cosine Approximation

Source:  Ted’s Math World, Calculator Magic #4

Retrieved 2014-10-13

cos x ≈ (((x^2/56 – 1)*x^2/30 + 1)*x^2/12 – 1)*x^2/2 + 1

Registers:  R0 = x in radians, R1 = cos x

 STEP KEY CODE STEP KEY CODE 1 STO 0 44, 0 17 1 1 2 2 2 18 + 40 3 Y^X 21 19 X<>Y 34 4 ENTER 36 20 1 1 5 ENTER 36 21 2 2 6 ENTER 36 22 ÷ 10 7 5 5 23 * 20 8 6 6 24 1 1 9 ÷ 10 25 - 30 10 1 1 26 X<>Y 34 11 - 30 27 2 2 12 X<>y 34 28 ÷ 10 13 3 3 29 * 20 14 0 0 30 1 1 15 ÷ 10 31 + 40 16 * 20 32 STO 1 44, 1 33 GTO 00 43,33,0

cos 0.33 ≈ 0.23847605343

cos -0.58 ≈ 0.83646264991

Rough Approximation for erf(x)

If x ≤ 1, accuracy is at least 3 digits.  Method:  3/8 Integration Rule

Source:  Ball, John A.   “Algorithms for RPN Calculators” John Wiley & Sons:  1978.

 STEP KEY CODE STEP KEY CODE 1 STO 0 44, 0 26 CHS 16 2 3 3 27 e^x 43, 22 3 ÷ 10 28 + 40 4 2 2 29 8 8 5 Y^X 21 30 ÷ 10 6 CHS 16 31 3 3 7 e^x 43, 22 32 * 20 8 ENTER 36 33 RCL 0 45, 0 9 RCL 0 45, 0 34 * 20 10 2 2 35 3 3 11 * 20 36 ÷ 10 12 3 3 37 1 1 13 ÷ 10 38 . 48 14 2 2 39 1 1 15 Y^X 21 40 2 2 16 CHS 16 41 8 8 17 e^x 43, 22 42 3 3 18 + 30 43 7 7 19 3 3 44 9 9 20 * 20 45 7 7 21 1 1 46 * 20 22 + 40 47 GTO 00 43,33,00 23 RCL 0 45, 0 24 2 2 25 Y^X 21

Examples:

x = .15;  Result ≈ 0.1680

x = .50;  Result ≈ 0.5205

x = 1.25; Result ≈ 0.9220  (actual ≈ 0.9229)

This blog is property of Edward Shore, 2014.