HP 15C: Error Function and Lower Tail Normal Cumulative Function

 HP 15C:   Error Function and Lower Tail Normal Cumulative Function

Formulas Used 

Error Function 

erf(x) = 2 ÷ √π * ∫( e^(-t^2) dt, t = 0 to t = x)

CDF (Cumulative Normal Distribution Function)

CDF(x) = 1 ÷ √(2 * π) * ∫( e^(-t^2) dt, t = -∞ to t = x)

We can use the error function to calculate CDF(x):

0.5 - erf(x), for x < 0

0.5 + erf(x), for x ≥ 0

This program uses the integration function inside the program.  We will need 23 free memory registers to run the program, which will be terminated at the end of program execution.

Labels used:

B:  erf function (error function)

C:  lower tail normal CDF 

4:  function used for integration

(Of course, feel free to use the labels you want, just be mindful to make the appropriate adjustments)

HP 15C Program Code:  Error Function and Lower Tail Normal Cumulative Function

Steps:  35

Bytes:  40

Step # :  Key Code :  Key

001 : 42,21,12:  LBL B

002 :  0  :   0

003 : 34 : x<>y

004 : 42,20, 4 : ∫_x_y 4

005 : 43, 32 : RTN

006 : 42,21,13 : LBL C

007 : 43,30, 2 :  TEST 2 (x<0)

008 : 43, 4, 0 : SF 0

009 : 43, 16 :  ABS

010 : 0   :   0

011 : 34 :  x<>y

012 : 2   :  2

013 : 11 :  √

014 : 10 :  ÷

015 : 42,20, 4 : ∫_y_x 4

016 : 2   :  2

017 : 10 :  ÷

018 : 48 :  .

019 : 5   :  5

020 : 34 :  x<>y

021 : 43, 6, 0 :  F?0

022 : 16  : CHS

023 : 40  :  +

024 : 43, 5, 0 : CF 0

025 : 43, 32 : RTN

026 : 42,21, 4 : LBL 4

027 : 43,11 : x^2

028 : 16 :  CHS

029 : 12 :  e^x

030 :  2  :  2

031 :  20  : ×

032 :  43,26 : π

033 :  11 :  √

034 :  10 :  ÷

035 :  43,32 : RTN

Examples

x = -0.7:  erf (N/A), CDF ≈ 0.2420

x = 0.7:  erf ≈ 0.6778, CDF ≈ 0.7580

x = 1:   erf ≈ 0.8427,  CDF ≈ 0.8413

x = 2.5  erf ≈ 0.9996,  CDF ≈ 0.9938

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Casio fx-3900PV: Program Bank

Casio fx-3900PV:  Program Bank

Here are a few programs for the Casio fx-3900Pv.  These programs should also work on the fx-3600P/fx-180P series (as long as the number of steps is less than 38) except that we can review and edit the steps on the fx-3900Pv.   See my review of the fx-3900Pv here:  

http://edspi31415.blogspot.com/2023/05/retro-review-casio-fx-3900pv.html

Error Function

The program allows the you to calculate the error function:

erf(b) = ∫( 2 * e^(-x²) ÷ √θ dx, x = 0, x = b)

Code:

001   Min

002   MR

003   x²

004   +/-

005   eˣ

006   ×

007   2

008   ÷

009   π

010   √

011   =

To calculate the erf(b):  run integral mode (Mode 1).  Select the program, enter n to get 2^n divisions by pressing [SHIFT] [RUN].   Next, enter 0 [RUN], then b [RUN].

Examples:  n = 6,  (2^6 = 64 divisions)

erf(1.5):

a = 0,  b = 1.5,  Result:  9.661051 * 10^-1

erf(0.4):

a = 0, b = 0.4,  Result:  4.28392 * 10^-1

erf(3.9):

a = 0, b = 3.9,  Result:  9.9999997 * 10^-1

Range and Height of a No-Air Projectile

R = v^2 * sin(2 * θ) ÷ g

H = v^2 * sin^2 θ ÷ (2 * g)

where:

R = range of the projectile (m)

H = maximum height of the projectile (m)

v = initial velocity (m/s)

θ = angle in degrees

g = Earth's gravity = 9.80665 m/s^2

Inputs:

K1 = v

K2 = θ

Outputs:

K3 = R

K4 = H

Used:

K5 = g

Code:

001   DEG  ( MODE 4 )

002   9

003   .

004   8

005   0

006   6

007   6

008   5

009   Kin 5

010   Kout 1

011   x²

012   ÷

013   Kout 5

014   ×

015   (

016   2

017   ×

018   Kout 2

019   )

020   SIN

021   =

022   Kin 3

023   HLT

024   Kout 1

025   x²

026   ×

027   Kout 2

028   SIN

029   x²

030   ÷

031   2

032   ÷

033   Kout 5

034   =

035   Kout 4

Examples:

Inputs:  K1 = 11.9 m/s,  K2 = 40°

Outputs:

R = 14.22082219 m

H = 2.98317166312 m

Inputs:  K1 = 11.9 m/s,  K2 = 60°

Outputs:

R = 12.0558115 m

H = 5.415075484 m

   

Magnitude and Phase of a LCR Series Circuit

Z = √(R² + (w * L - 1 ÷ (w * C))² )

θ = arctan((w * L - 1 ÷ (w * C)) ÷ R)

where:

Z = magnitude (Ω)

θ = phase angle (degrees)

R = resistance (Ω)

L = inductance (H)

C = capacitance (F)

w = angular frequency, where w = 2 * π * f

f = frequency

Inputs:  R, f, C, L

M = f  

K2 = L

K3 = R

K4 = C

Outputs:

K1 = w

K5 = Z

K6 = θ

Note that the source (see Source section below) omitted the square root in the formula for Z.  The above formula is correct.

Code:

001   DEG    (Mode 4)

002   MR

003   Kin 1

004   2

005   Kin× 1   (shown as K×1)

006   π

007   Kin× 1    (shown as K×1)

008   Kout 1

009   ×

010   Kout 2

011   -

012   (

013   Kout 1

014   ×

015   Kout 4

016   )

017   1/x

018   =

019   Kin 6

020   Kout 3

021   x²

022   +

023   Kout 6

024   x²

025   =

026   √

027   Kin 5

028   HLT

029   (

030   Kout 6

031   ÷

032   Kout 3

033   )

034   tan⁻¹

035   =

036   Kin 6

Example:

Inputs:

f = 100 Hz,  [ Min ]

L = 4 * 10^-3 H  (store in K2)

R = 6300 Ω  (store in K3)

C =  5 * 10^-6 F  (store in K4)

Outputs:

Z = 6307.909915 Ω  (K5)

θ = -2.869631956° (K6)

Quadratic Equations:  Real Roots

Solve the equation: 

x^2 + K1 * x + K2 = 0

The roots are:

x = (-K1 ± √(K1^2 - 4 * K2)) ÷ 2

Inputs:

K1 = coefficient of x

K2 = constant

Outputs:

K4 = root 1

K5 = root 2

Used:  K3 = √(K1^2 - 4 * K2)

Code:

001    (

002    Kout 1

003    x²

004    -

005    4    

006    ×

007    Kout 2

008    )

009    √

010    Kin 3

011    (

012    Kout 3

013    +/-

014    -

015    Kout 1

016    )

017    ÷

018    2

019    =

020    Kin 4

021    HLT

022    +

023    Kout 3

024    =

025    Kin 5

Example:

Solve x^2 - 19 * x + 10 = 0

Inputs:

K1 = -19

K2 = 90

Outputs:  9, 10

Source 

(for Range and Height of a No-Air Projectile and Magnitude and Phase of a LCR Series Circuit):

Rosenstein, Morton.   Computing With the Scientific Calculator.  Casio.  1986

Enjoy,

Eddie 

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP 42S Programming Part I: Matrix Column Sum, GCD, Error Function

HP 42S Programming Part I:  Matrix Column Sum, GCD, Error Function

Continuing with calculators from the 1980s, this series will present one of the most beloved scientific calculators and one of the most sought after: the Hewlett Packard HP 42S from the late 1980s.  If you want one, you might want to save up because HP 42S calculators are valuable, and often cost over $100.

It is one of my most favorite calculators of all time. 

Free42

However, you can use an emulator for the HP 42S for free, check out the Free42 by Thomas Oakken.  It is a rewrite of the HP 42S code and the emulator is available on many platforms including Widows, Mac, iOS, and Android.  Link:  http://thomasokken.com/free42/   

Oakken accepts donations.

Link to Part II: Dew Point, Ellipse Area and Eccentricity, Easy Transverse

Part III hopefully to come next week.  

As usual, anything that comes after the double-backslash is a comment only.

HP 42S: Column Sums of a Matrix

This could potentially had been a medium to long sized programs involving loops, but thanks to the powerful function RSUM (found in CATALOG-FUNC or executed by XEQ RSUM) this program will be quite short.  RSUM is the row sum function, or the sum of each row of a matrix.

Just put a matrix on the stack and run CSUM.  When the program is finished, press [shift] [ 9 ] (MATRIX), {EDIT} to see the sums.

00 {12-Byte Prgm}

01 LBL “CSUM”

02 TRANS \\ transpose

03 RSUM

04 END

Test:  MAT1 = [ [4, 3, 1], [5, 8, 2], [7, 6, 3]]

Result:  [16, 17, 6]

HP 42S:  GCD (Greatest Common Divisor) by Euclid Division

Enter the two integers on the Y stack and X stack.  Order doesn’t matter.

00 {42-Byte Prgm}

01 LBL “GCD”

02 X<Y?

03 X<>Y

04 STO 01  \\ store maximum in R01

05 X<>Y

06 STO 00  \\ store minimum in R00

07 LBL 00 \\ main loop

08 RCL 01

09 ENTER

10 RCL÷ 00

11 IP

12 RCL* 00

13 –

14 X=0?  \\ is max – IP(min/max)*min = 0?

15 GTO 01

16 X<>Y

17 STO 01

18 X<>Y

19 STO 00

20 GTO 00

21 LBL 01 \\ display GCD

22 RCL 00

23 “GCD:”

24 ARCL ST X

25 AVIEW

26 END

Test:   Find the GCD of 485 and 175.

485 [ENTER] 175 [XEQ] {GCD}   Result:  5

Alternatively:

175 [ENTER] 485 [XEQ] {GCD}   Result:  5

HP 42S:  Error (erf) Function

ERF(X) = ∫ 2/√π * (e^(-T^2)) dT, from T = 0 to T = x

Integrand:

00 {22-Byte Prgm}

01 LBL “ERF”

02 MVAR “T”

03 RCL “T”

04 X^2

05 +/-

06 E^X

07 2

08 *

09 PI

10 SQRT

11 ÷

12 END

Instructions:

Press [(shift)] [ 8 ] (∫ f(x) )

Choose ERF at the “Select ∫f(x) Program”

Select T at Set Vars; Select ∫var

Store 0 in LLIM, X in ULIM, a suitable accuracy factor (10^-n) in ACC

Press the ∫ soft key.

Test: erf(1.2) with ACC of 1E-6  (LLIM = 0, ULIM = 1.2) 

Result:  0.9103

 This blog is property of Edward Shore, 2016.

TI-55 III Programs Part II: Impedance of a Series RLC Circuit, Quadratic Equation, Error Function

TI-55 III Programs Part II: Impedance of a Series RLC Circuit, Quadratic Equation, Error Function

On to Part II!

For Part I, click here:  Digital Root, Complex Number Multiplication, Escape Velocity

For Part III, click here: Area and Eccentricity of Ellipses, Determinant and Inverse of 2x2 Matrices, Speed of Sound/Principal Frequency

TI-55 III:  Impedance of Series RLC Circuit

The impedance of a series RLC circuit in Ω (ohms) is:

Z = √(R^2 + (2*π*f*L – 1/(2*π*f*C))^2)

Where:

R = resistance of the resistor in ohms (Ω)

L = inductance of the inductor in Henrys (H)

C = capacitance of the capacitor in Farads (F)

f = resonance frequency in Hertz (Hz)

XL = 2*π*f*L

XC = 1/(2*π*f*C)

Program:

Partitions allowed: 1-4

STEP	CODE	KEY	COMMENT
00	65	*	Start with f
01	02	2
02	65	*
03	91	π
04	95	=
05	61	STO
06	00	0	Store 2πf in R0
07	65	*
08	12	R/S	Prompt for L
09	75	-
10	53	(
11	71	RCL
12	00	0
13	65	*
14	12	R/S	Prompt for C
15	54	)
16	17	1/x
18	18	X^2
19	85	+
20	12	R/S	Prompt for R
21	18	X^2
22	95	=
23	13	√
24	41	INV
25	47	Eng	Cancel Eng Notation
26	12	R/S	Display Z

Input:  f [RST] [R/S], L [R/S], C [R/S], R [R/S]

Result:  Z

Test:

f = 60 Hz

L = 0.25 H

C = 16 * 10^-6 F

R = 150 Ω

Result:  166.18600 Ω

TI-55 III: Quadratic Equation

This program find the real roots of the equation:

X^2 + B*X + C = 0

Where:

D = B^2 – 4*C

If D ≥ 0, then continue the program since it will find the real roots.  Otherwise, stop since the roots are complex and is beyond the scope of this program.  The two roots are:

X1 = (-B + √D)/2

X2 = (-B - √D)/2

Program:

Partitions Allowed: 3

STEP	CODE	KEY	COMMENT
00	71	RCL	Calculate Discriminant
01	00	0
02	18	X^2
03	75	-
04	04	4
05	65	*
06	71	RCL
07	01	1
08	95	=
09	12	R/S	Display Discriminant
10	13	√
11	61	STO
12	02	2
13	75	-
14	71	RCL
15	00	0
16	95	=
17	55	÷
18	02	2
19	95	=
20	12	R/S	Display X1
21	53	(
22	71	RCL
23	00	0
24	85	+
25	71	RCL
26	02	2
27	54	)
28	94	+/-
29	55	÷
30	02	2
31	95	=
32	12	R/S	Display X2

Input:  B [STO] 0, C [STO] 1, [RST] [R/S]

Results:  Discriminant [R/S], root 1 [R/S], root 2

Test:  Solve X^2 + 0.05*X – 1 = 0

Input:  0.05 [STO] 0, 1 [+/-] [STO] 1 [RST] [R/S]

Results:  Discriminant = 4.0025  (It is non-negative, continue) [R/S]

X1 ≈ 0.9753125  [R/S]

X2 ≈ -1.0253125

TI-55 III: Gaussian Error Function

The error function is defined as:

erf(x) = ∫( 2*e^(-t^2)/√π dt, from t = 0 to t = x)

This program illustrates the integration function [ ∫ dx ] on the TI-55 III.

Program:

Prepare by pressing [2^nd] [LRN] (Part) 3.  Integration needs a minimum of 3 memory registers.  That means, f(x) can take a maximum of 40 steps.

STEP	CODE	KEY	COMMENT
00	18	X^2	Integrand
01	94	+/-
02	41	INV
03	44	ln x	[INV] [ln x]: e^x (EXP)
04	65	*
05	02	2
06	55	÷
07	91	π
08	13	√
09	95	=
10	12	R/S
11	22	RST	End f(x) with =,R/S,RST

Input:  0 [STO] 1 (lower limit), x [STO] 2 (upper limit), [ ∫ dx ] n (number of partitions) [R/S]

Result: erf(x)

Test 1: erf(1.2) ≈ 0.910314. I use 12 partitions.

Input:  0 [STO] 1, 1.2 [STO] 2, [ ∫ dx ] 12 [ R/S ]

Result:  0.910314

Test 2:  erf(0.9) ≈ 0.7969082.   Store 0 in R0, 0.9 in R1.  12 partitions are used.

 Eddie

This blog is property of Edward Shore, 2016.