ToolkitP - TVM, NewtonP, Stat5P, CurveP, StrayLightP - J E Patterson

J E Patterson - jepspectro.com - 20231119

Description

This Toolkit is intended to help a little with the development of scientific equipment and processing the data produced. If income is earned the Toolkit can help with that too.

The program will run on Torsten Manz's hp-15c Simulator and SwissMicros DM15 calculator. The program is large at over 500 program steps, so extended memory needs to be set up.
In the hp-15c Simulator Preferences, under DM-15, set the number of registers to 229.
For the SwissMicros DM15, install the M1B extended memory version of the DM15 firmware.
This is the default when purchased.

Right clicking on GSB will display a data entry guide for each program. The indented entries are program subroutines. Print the following Toolkit Data Entry as a compact reminder.

Toolkit Data Entry

TVM n→1 i→2 PV→3 PMT→4 FV→5 B/E→6 X→I f SOLVE A
NewtonP Enter one guess, then GSB B
CurveP y1 ENTER y2 ENTER y3 GSB C, x1 ENTER x2 ENTER x3 R/S
x' GSB D, y' is displayed, repeat
Stat5P Data Entry, y ENTER x GSB 1, repeat
GSB 2 Linear y=ax+b, b, a and r are displayed. Enter x' R/S, y' is displayed, repeat
GSB 3 Log y=a*ln(x)+b, b, a and r are displayed. Enter x' R/S, y' is displayed, repeat
GSB 4 Power y=x^a*b,b, a and r are displayed. Enter x' R/S, y' is displayed, repeat
GSB 5 Exp y=e^ax*b, b, a and r are displayed. Enter x' R/S, y' is displayed, repeat
Stray Light y1 ENTER y2 GSB 6, x1 ENTER x2 R/S. Enter x' GSB 7, y' is displayed, repeat

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TVM - Time Value of Money - Accurate TVM usage instructions - Jeff Kearns

TVM starts at line 1 and finishes at line 40.

1. Store 4 of the following 5 variables, using appropriate cash flow conventions as follows:

n STO 1 --- Number of compounding periods
i STO 2 --- Interest rate (periodic) expressed as a %
PV STO 3 --- Present Value
PMT STO 4 --- Periodic Payment
FV STO 5 --- Future Value

Store the appropriate value (1 for Annuity Due or 0 for Regular Annuity) as
B/E STO 6 --- Begin/End Mode. The default is 0 for regular annuity or End Mode.

2. Store the register number X containing the floating variable to the indirect storage register as X STO I.

3. f SOLVE A

Example from the HP-15C Advanced Functions Handbook - Page 145 and Page 151

"Many Pennies (alternatively known as A Penny for Your Thoughts):

A corporation retains Susan as a scientific and engineering consultant.
Her fee is one penny per second for her thoughts, paid every second of every day for a year.
Rather than distract her with the sounds of pennies dropping, the corporation proposes to deposit them for her into a bank account.
Interest accrues at the rate of 11.25 percent per annum compounded every second.
At year's end these pennies will accumulate to a sum

total = (payment) x ((1+i/n)^n-1)/(i/n)

where payment = -$0.01 = one penny per second. Note that, by convention, payment is negative and income is positive.
i = 0.1125 = 11.25 percent per annum interest rate,
n = 60 x 60 x 24 x 365 = number of seconds in a year.

Using her HP-15C, Susan reckons that the total will be $376,877.67.
But at year's end the bank account is found to hold $333,783.35 .
Is Susan entitled to the $43,094.32 difference?"

31,536,000 STO 1
(11.25/31,536,000) STO 2
0 STO 3
-0.01 STO 4
5 STO I
f SOLVE A

The HP-15C now gives the correct result: $333,783.35.

Thanks to Thomas Klemm for debugging the routine.
The code has been edited to reflect Thomas' suggested changes.

Jeff Kearns

The hyperbolic sine in this program calculates e^x-1 more accurately using 2.sinh(x/2).e^x/2 from this reference by Thomas Klemm (Post #14).

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NewtonP - Newton-Raphson equation solver - J E Patterson

NewtonP starts at line 41 and finishes at line 68.

GSB B runs a Newton-Raphson solver which acts on f(x) = 0 under label E. Only one guess is required.
SOLVE the internal solver can also be run using two guesses.

Label E can hold any equation of the form f(x) = 0. Note that some of the hp-15C Owner's Handbook examples may require some extra ENTER statements at the beginning as the stack is expected to be filled with x. The open box example on page 189 requires three ENTER statements at the beginning. This is the default example under label E.

Instructions

A problem from the hp-15c Owner's Handbook - page 189:

A 4 decimetre by 8 decimetre metal sheet is available, i.e. 400 mm by 800 mm.
The box should hold a volume V of 7.5 cubic decimetres, i.e. 7.5 litres.
How should the metal be folded for the tallest box in decimetres.
We are using decimetres rather than mm because equation entry is simplified.
Let x be the height.
Volume V = (8 - 2x)(4-2x)x. There are two sides and two ends of height x.
Rearrange to f(x) = 4((x - 6)x + 8)x - V = 0 and solve for the height x.

There are 3 solutions depending on the guesses.
The initial guess can be recovered with RCL 1

0 GSB B gives x = 0.2974 decimetres or 29.74 mm - a flat box
1 GSB B gives x = 1.5 decimetres or 150 mm - a reasonable height box.
2 GSB B gives x = 1.5 decimetres or 150 mm - a reasonable height box.
3 GSB B gives x = 0.2974 decimetres or 29.74 mm - a flat box. Here the secant line now intersects the x axis below the smallest root.
4 GSB B gives x = 4.2026 decimetres or 420.26 mm - an impossible box.

Solver

Choose a starting point x1
do
Calculate f(x1)
If x1 = 0 use 1 instead to avoid a divide by zero from the derivative f'(x1)
define h = 1/10000
calculate f'(x1) ≈ (f(x1+h) - f(x1))/h
calculate f(x1)/f'(x1)
subtract from x1
Loop until f(x1) = 0 within an error tolerance determined by the FIX, SCI or ENG significant digits setting.
Display root x1.

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CurveP - Fit calibration curves - J E Patterson

CurveP starts at line 69 and finishes at line 221.

The program fits data that may be linear at low x,y amplitudes but y falls off at higher x values. This is a common issue in many physical systems. x1, y1 should preferably be on the linear part of the curve. CurveP is not a regression. It assumes that the data is reasonably precise. Linear and power fit data sets also work well.

The equation used is:

y = a.x^b + c.x.e^-d.x

See Curve Fitting at jepspectro.com.

Originally chart recorders were used to obtain data which occupied the y axis, leaving interpreted results naturally plotted on the x axis, when graphed on the same paper. Here we are using y as the equation output axis.

The parameters a = scale, b = order, c = slope and d = factor are calulated by iteration using 3 standards and a blank. x is first normalised, the equation solved and un-normalised to get y. This limits the magnitude of calculated values in the iteration.

y1, y2 and y3 are known standards with respective readings of x1, x2 and x3.

y1 ENTER, y2 ENTER, y3 GSB C
x1 ENTER, x2 ENTER, x3 R/S

Do not press ENTER after y3 or x3.
Note that y3>y2>y1 and x3>x2>x1.
After normalisation y3 = x3 = 10 so x3 is not saved. y3 is used instead in the program.

The curve runs through the origin and three points (x1,y1) (x2,y2) (x3,y3). The order of the upper part of the curve is displayed.

Enter x GSB D to get y.

If there is a blank value, enter it as data and subtract the output from all the results.

Test [x,y] data sets

[2,2 4,5 5,8] [1,1 2,4 3,9] [1,1 2,2 3,3] [20,3.69 30,8.64 47,22.16]
The last data set relates X-ray fluorescence k line energy in keV to the element atomic number - see Moseley's Law.

Notes:

In this update I have improved the starting values for the iteration. Order is limited to 10.

Because only 3 standards and a blank are used a regression approach is not suitable. The iteration attempts to fit the standards and blank exactly. The standards need to be accurately prepared. The calibration curve can be validated by running other standards as samples.

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Stat5P - Linear, logarithmic, power and exponential regressions - J E Patterson

Stat5P starts at line 222 and finishes at line 423.

This program provides linear, logarithmic, power and exponential regressions with a single entry of a data set.

This program is an adaptation of software I wrote for a personal computer in 1984. For most statistical work a personal computer is best, but this program is good for a first look at experimental data.

Introduction:

Entered numbers should be positive.
Enter 1 EEX 9 CHS in place of zero.
For linear regressions the built-in HP-15 linear-regression function can be used if a wider number range is required.
The most recent data pair can be deleted with GSB .6. The statistics counter n is also decremented by 1.
GSB 9 - zeros registers and clears flag 1.
If there is an obvious inverse relationship between x and y press GSB .7 which sets flag 1 to 1. This inverts any x data entered.

y ENTER x GSB 1
Repeat to enter more data.

GSB 2 for linear regression y = ax+b
GSB 3 for logarithmic regression y = a ln(x) + b
GSB 4 for power regression y = x^a * b
GSB 5 for exponential regression y = e^ax * b

On pressing any of these keys b is first displayed, then a and finally the correlation coefficient r is left in the display.

x' R/S displays a y' estimate.

GSB 2, 3, 4, and 5 can be repeated at any time without entering new data.

Example:

y = 2, 4, 5, 6
x = 2, 5, 8, 13

GSB 9 to clear registers
2 ENTER 2 GSB 1
4 ENTER 5 GSB 1
5 ENTER 8 GSB 1
6 ENTER 13 GSB 1

GSB 2
Linear regression y = ax+b
b = 1.8106
a = 0.3485
r = 0.9571

GSB 3
Logarithmic regression y = a ln(x) + b
b = 0.5318
a = 2.1409
r = 0.9999

GSB 4
Power regression y = x^a * b
b = 1.4005
a = 0.5950
r = 0.9870

GSB 5
Exponential regression y = e^ax * b
b = 2.0563
a = 0.0928
r = 0.9048

Clearly a logarithmic fit to the data is reasonable because the correlation coefficient r is closest to 1.
Once r is displayed x' can be entered to estimate y'
Note that a can also be recovered with RCL . 1 and b with RCL . 0.

5 R/S displays 3.9774 as the y' estimate instead of the original y entry of 4.
Repeat for more values of x'. This applies for all regression models.
Note that even with r = 0.9999 the fit is not quite perfect. To 6 places r = 0.999911.

It is easy to compare different regression models. Just enter GSB 2 to GSB 5 as often as required to show b, a and r for the different regression models and then enter x' R/S to estimate y'.

Test data sets

Linear y = ax+b
x = 1, 2, 3, 4
y = 7, 9, 11, 13
b = 5.0000
a = 2.0000
r = 1.0000
3 R/S displays 11.0000

Logarithmic y = a ln(x) + b
x = 1, 2, 3, 4
y = 5, 6.3863, 7.1972, 7.7726
b = 5.0000
a = 2.0000
r = 1.0000
3 R/S displays 7.1972.

Power y = x^a * b
x = 1, 2, 3, 4
y = 5, 20, 45, 80
b = 5.0000
a = 2.0000
r = 1.0000
3 R/S displays 45.0000.

Exponential y = e^ax * b
x = 1, 2, 3, 4
y = 36.9453, 272.9908, 2017.1440, 14904.7899
b = 5.0000
a = 2.0000
r = 1.0000
3 R/S displays 2017.1442.

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StrayLightP - Stray light curve correction - J E Patterson

StrayLightP starts at line 424 and finishes at line 499.

Spectrophotometers have some stray light or detector dark current which causes the calibration to be non linear at high absorbances.

Absorbance = Log (Incident light Intensity)/(Transmitted light Intensity).

This program corrects for the calibration curvature caused by the presence of stray light.

Absorbance A = log(Io/I) .... ideal
Io is the incident light intensity and I is the transmitted light intensity.
A' = log(Io+Is)/(I+Is) .... measured absorbance
Is is the stray light intensity (or the dark current in the detector)
A = A'-log(k(1-10^A')+1 ....................................................... (1)
where k = Is/Io and 100k = % stray light
k = (10^(A'-A)-1)/(1-10^A')
C1 and C2 are standards and A1' and A2' are the measured absorbances.
k2 = (10^(A2'-A2)-1)/(1-10^A2') ............................................... (2)
Substituting A1 in equation (1)
A1 = A1'-log(k1(1-10^A1')+1)
For ideal absorbances A2 = A1(C2/C1)
A2 = (A1'-log(k1((1-10^A1')+1)C2/C1 ................................ (3)

Equation (3) can be substituted in equation (2) and an iteration runs until (k1 - k2)*100 = 0 within an error tolerance determined by the FIX, SCI or ENG significant digits setting.
The value for k is now set equal to k1, the last value calculated.
100k = % stray light

Input two standard concentration values C1 ENTER C2 GSB 6.
Input two corresponding absorbances A1' ENTER A2' R/S.
The % stray light is displayed. This is stored in register 8.

Input an unknown sample absorbance A' R/S the resulting concentration C' is displayed and stored in register 9.
Repeat.

Example 1:
Standard data: (Concentrations C1, C2 = 5, 10, Absorbances A1', A2' = 0.53, 0.83).
5 ENTER
1 0 GSB 6
. 5 3 ENTER
. 8 7 R/S
Stray light is 9.62%

This curvature is not unusual for Atomic Absorption spectrometry. Spectrophotometry can be much more linear in the near absence of stray light.

Example 2:
Unknown sample absorbance is 0.74
. 7 4 R/S
The unknown sample concentration is 7.66 units, using the same concentration units as the standards.
Repeat A' R/S for further samples.

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Program Resources

Labels

Name	Description	Name	Description	Name	Description
A	TVM n→1 i→2 PV→3 PMT→4 FV→5 B/E→6 X→I Solve A	4	Stat5P Power y=x^a*b, R/S=a, R/S=r, x'R/S=y',	.3	Stat5P Logarithmic, estimate y' from x' loop
B	NewtonP Enter one guess, then GSB B	5	Stat5P Exp y=e^ax*b, R/S=a, R/S=r, x'R/S=y'	.4	Stat5P, Power, estimate y' from x' loop
C	CurveP y1 ENTER y2 ENTER y3 GSB C, x1 ENTER x2 ENTER x3 R/S	6	Stray Light y1 ENTER y2 GSB 6, x1 ENTER x2 R/S	.5	Stat5P Exponential, estimate y' from x' loop
D	CurveP x' GSB D=y', repeat	7	Stray Light x'GSB 7=y', repeat	.6	Stat5P Delete most recent data pair, decrement n by 1
E	NewtonP or Solve - Equation to be solved	8	NewtonP loop and test	.7	Stat5P Set flag 1 to 1
0	Stray Light iteration	9	Stat5P Initialise - zero registers, set flag 1 to zero	.8	CurveP Loop
1	Stat5P Data Entry, y ENTER x GSB 1, repeat	.0	Stat5P Linear or Logarithmic regression subroutine	.9	CurveP RCL.5 1/x STO.5
2	Stat5P Linear y=ax+b, R/S=a, R/S=r, x'R/S=y'	.1	Stat5P Power or exponential regression subroutine
3	Stat5P Log y=a*ln(x)+b, R/S=a, R/S=r, x'R/S=y'	.2	Stat5P Linear, estimate y' from x' loop

Storage Registers

Name	Description	Name	Description	Name	Description
0	∑x, Fraction decimal input value	7	∑xy" copied from appropriate registers, xscale, temp 2	.4	∑(lnx)^2, a - scale
1	∑x^2, y1, N, Result ,Guess 1, X1 conc	8	∑y, yscale, % stray light	.5	∑lny, temp1
2	n, y2, I, X2 conc	9	∑y^2, c - slope	.6	∑(lny)^2, temp2
3	∑x" copied from appropriate registers, y3	.0	Temp x and b, absolute decimal number for fraction	.7	∑((lnx) * (Lny)), temp3
4	∑x^2" copied from appropriate registers, b-order, PMT, stray light fraction.	.1	Temporary y and a storage, x1, y1 abs	.8	∑(x * lny), xinput
5	∑y" copied from appropriate registers, d-factor, FV, stray light old	.2	∑xy, x2, y2 abs	.9	∑(lnx *y)
6	∑y^2" copied from appropriate registers,count, B/E 1/0, temp 1	.3	∑lnx, b - order first guess, yabs	(i)	Register number of TVM value required

Flags

Number	Description
1	Invert x data flag

Program

Line	Display	Key Sequence	Line	Display	Key Sequence	Line	Display	Key Sequence
000			173	10	÷	346	12	eˣ
001	42,21,11	f LBL A	174	15	1/x	347	45 .0	RCL . 0
002	44 24	STO (i)	175	1	1	348	20	×
003	45 2	RCL 2	176	40	+	349	22 .5	GTO . 5
004	26	EEX	177	44 .7	STO . 7	350	43 32	g RTN
005	2	2	178	45,20, 4	RCL × 4	351	42,21, 9	f LBL 9
006	10	÷	179	44 4	STO 4	352	42 34	f REG
007	36	ENTER	180	45 .7	RCL . 7	353	43 35	g CLx
008	36	ENTER	181	1	1	354	43 32	g RTN
009	1	1	182	30	−	355	42,21, .0	f LBL . 0
010	40	+	183	43 16	g ABS	356	42 49	f L.R.
011	43 12	g LN	184	1	1	357	44 .0	STO . 0
012	34	x↔y	185	26	EEX	358	42 31	f PSE
013	43 36	g LSTx	186	16	CHS	359	34	x↔y
014	1	1	187	6	6	360	44 .1	STO . 1
015	43,30, 6	g TEST x≠y	188	43,30, 8	g TEST x<y	361	42 31	f PSE
016	30	−	189	22 .8	GTO . 8	362	1	1
017	10	÷	190	45 4	RCL 4	363	42 48	f ŷ,r
018	20	×	191	43 32	g RTN	364	34	x↔y
019	45,20, 1	RCL × 1	192	42,21,14	f LBL D	365	43 32	g RTN
020	36	ENTER	193	44 .8	STO . 8	366	42,21, .1	f LBL . 1
021	12	eˣ	194	45,10, 7	RCL ÷ 7	367	42 49	f L.R.
022	45,20, 3	RCL × 3	195	45 3	RCL 3	368	12	eˣ
023	34	x↔y	196	20	×	369	44 .0	STO . 0
024	2	2	197	44 .8	STO . 8	370	42 31	f PSE
025	10	÷	198	45 4	RCL 4	371	34	x↔y
026	42,22,23	f HYP SIN	199	14	yˣ	372	44 .1	STO . 1
027	43 36	g LSTx	200	45,20, .4	RCL × . 4	373	42 31	f PSE
028	12	eˣ	201	45 5	RCL 5	374	1	1
029	20	×	202	45,20, .8	RCL × . 8	375	42 48	f ŷ,r
030	2	2	203	16	CHS	376	34	x↔y
031	20	×	204	12	eˣ	377	43 32	g RTN
032	45,20, 4	RCL × 4	205	45,20, .8	RCL × . 8	378	42,21, .6	f LBL . 6
033	26	EEX	206	45,20, 9	RCL × 9	379	45 .1	RCL . 1
034	2	2	207	40	+	380	44,30, 8	STO − 8
035	45,10, 2	RCL ÷ 2	208	45,20, 8	RCL × 8	381	43 11	g x²
036	45,40, 6	RCL + 6	209	45 3	RCL 3	382	44,30, 9	STO − 9
037	20	×	210	10	÷	383	11	√x̅
038	40	+	211	43 32	g RTN	384	43 12	g LN
039	45,40, 5	RCL + 5	212	42,21, .9	f LBL . 9	385	44,30, .5	STO − . 5
040	43 32	g RTN	213	45 .5	RCL . 5	386	43 11	g x²
041	42,21,12	f LBL B	214	15	1/x	387	44,30, .6	STO − . 6
042	44 .1	STO . 1	215	44 .5	STO . 5	388	45 .0	RCL . 0
043	44 1	STO 1	216	43 32	g RTN	389	44,30, 0	STO − 0
044	42,21, 8	f LBL 8	217	42,21, 9	f LBL 9	390	43 11	g x²
045	45 .1	RCL . 1	218	42 34	f REG	391	44,30, 1	STO − 1
046	32 15	GSB E	219	43, 5, 1	g CF 1	392	11	√x̅
047	44 .4	STO . 4	220	43 35	g CLx	393	43 12	g LN
048	45 .1	RCL . 1	221	43 32	g RTN	394	44,30, .3	STO − . 3
049	43 20	g x=0	222	42,21, 1	f LBL 1	395	43 11	g x²
050	12	eˣ	223	43, 6, 1	g F? 1	396	44,30, .4	STO − . 4
051	26	EEX	224	15	1/x	397	45 .0	RCL . 0
052	4	4	225	44 .0	STO . 0	398	45 .1	RCL . 1
053	10	÷	226	34	x↔y	399	20	×
054	44 .2	STO . 2	227	44 .1	STO . 1	400	44,30, .2	STO − . 2
055	45,40, .1	RCL + . 1	228	44,40, 8	STO + 8	401	45 .0	RCL . 0
056	32 15	GSB E	229	43 11	g x²	402	45 .1	RCL . 1
057	45,30, .4	RCL − . 4	230	44,40, 9	STO + 9	403	43 12	g LN
058	45,10, .2	RCL ÷ . 2	231	11	√x̅	404	20	×
059	45 .4	RCL . 4	232	43 12	g LN	405	44,30, .8	STO − . 8
060	34	x↔y	233	44,40, .5	STO + . 5	406	45 .0	RCL . 0
061	10	÷	234	43 11	g x²	407	43 12	g LN
062	44,30, .1	STO − . 1	235	44,40, .6	STO + . 6	408	45 .1	RCL . 1
063	45 .4	RCL . 4	236	45 .0	RCL . 0	409	20	×
064	43 34	g RND	237	44,40, 0	STO + 0	410	44,30, .9	STO − . 9
065	43,30, 0	g TEST x≠0	238	43 11	g x²	411	45 .0	RCL . 0
066	22 8	GTO 8	239	44,40, 1	STO + 1	412	43 12	g LN
067	45 .1	RCL . 1	240	11	√x̅	413	45 .1	RCL . 1
068	43 32	g RTN	241	43 12	g LN	414	43 12	g LN
069	42,21,13	f LBL C	242	44,40, .3	STO + . 3	415	20	×
070	44 8	STO 8	243	43 11	g x²	416	44,30, .7	STO − . 7
071	33	R⬇	244	44,40, .4	STO + . 4	417	1	1
072	44 2	STO 2	245	45 .0	RCL . 0	418	44,30, 2	STO − 2
073	33	R⬇	246	45 .1	RCL . 1	419	45 2	RCL 2
074	44 1	STO 1	247	20	×	420	43 32	g RTN
075	45 8	RCL 8	248	44,40, .2	STO + . 2	421	42,21, .7	f LBL . 7
076	1	1	249	45 .0	RCL . 0	422	43, 4, 1	g SF 1
077	0	0	250	45 .1	RCL . 1	423	43 32	g RTN
078	44 3	STO 3	251	43 12	g LN	424	42,21, 6	f LBL 6
079	10	÷	252	20	×	425	44 2	STO 2
080	44,10, 2	STO ÷ 2	253	44,40, .8	STO + . 8	426	33	R⬇
081	44,10, 1	STO ÷ 1	254	45 .0	RCL . 0	427	44 1	STO 1
082	31	R/S	255	43 12	g LN	428	31	R/S
083	44 7	STO 7	256	45 .1	RCL . 1	429	44 .2	STO . 2
084	33	R⬇	257	20	×	430	33	R⬇
085	44 .2	STO . 2	258	44,40, .9	STO + . 9	431	44 .1	STO . 1
086	33	R⬇	259	45 .0	RCL . 0	432	0	0
087	44 .1	STO . 1	260	43 12	g LN	433	44 4	STO 4
088	45 7	RCL 7	261	45 .1	RCL . 1	434	42,21, 0	f LBL 0
089	45 3	RCL 3	262	43 12	g LN	435	45 4	RCL 4
090	10	÷	263	20	×	436	44 5	STO 5
091	44,10, .2	STO ÷ . 2	264	44,40, .7	STO + . 7	437	45 .1	RCL . 1
092	44,10, .1	STO ÷ . 1	265	1	1	438	1	1
093	45 3	RCL 3	266	44,40, 2	STO + 2	439	45 .1	RCL . 1
094	45 2	RCL 2	267	45 2	RCL 2	440	13	10ˣ
095	10	÷	268	43 32	g RTN	441	30	−
096	43 12	g LN	269	42,21, 2	f LBL 2	442	45 5	RCL 5
097	45 3	RCL 3	270	45 0	RCL 0	443	20	×
098	45 .2	RCL . 2	271	44 3	STO 3	444	1	1
099	10	÷	272	45 8	RCL 8	445	40	+
100	43 12	g LN	273	44 5	STO 5	446	43 13	g LOG
101	10	÷	274	45 1	RCL 1	447	30	−
102	44 4	STO 4	275	44 4	STO 4	448	44 6	STO 6
103	44 .3	STO . 3	276	45 9	RCL 9	449	45 2	RCL 2
104	1	1	277	44 6	STO 6	450	45 1	RCL 1
105	44 5	STO 5	278	45 .2	RCL . 2	451	10	÷
106	0	0	279	44 7	STO 7	452	20	×
107	44 6	STO 6	280	32 .0	GSB . 0	453	44 7	STO 7
108	45 1	RCL 1	281	42,21, .2	f LBL . 2	454	45 .2	RCL . 2
109	45,10, .1	RCL ÷ . 1	282	31	R/S	455	45 7	RCL 7
110	44 9	STO 9	283	45 .1	RCL . 1	456	30	−
111	42,21, .8	f LBL . 8	284	20	×	457	13	10ˣ
112	1	1	285	45 .0	RCL . 0	458	1	1
113	44,40, 6	STO + 6	286	40	+	459	30	−
114	45 3	RCL 3	287	22 .2	GTO . 2	460	1	1
115	45 4	RCL 4	288	43 32	g RTN	461	36	ENTER
116	43,30, 7	g TEST x>y	289	42,21, 3	f LBL 3	462	45 .2	RCL . 2
117	45 .3	RCL . 3	290	45 .3	RCL . 3	463	13	10ˣ
118	44 4	STO 4	291	44 3	STO 3	464	30	−
119	45 3	RCL 3	292	45 .4	RCL . 4	465	10	÷
120	45 5	RCL 5	293	44 4	STO 4	466	44 4	STO 4
121	45,20, 3	RCL × 3	294	45 8	RCL 8	467	45 5	RCL 5
122	16	CHS	295	44 5	STO 5	468	30	−
123	12	eˣ	296	45 9	RCL 9	469	26	EEX
124	45,20, 3	RCL × 3	297	44 6	STO 6	470	2	2
125	45,20, 9	RCL × 9	298	45 .9	RCL . 9	471	20	×
126	30	−	299	44 7	STO 7	472	43 34	g RND
127	45 3	RCL 3	300	32 .0	GSB . 0	473	43,30, 0	g TEST x≠0
128	45 4	RCL 4	301	42,21, .3	f LBL . 3	474	22 0	GTO 0
129	43,30, 2	g TEST x<0	302	31	R/S	475	45 4	RCL 4
130	45 .3	RCL . 3	303	43 12	g LN	476	26	EEX
131	44 4	STO 4	304	45 .1	RCL . 1	477	2	2
132	14	yˣ	305	20	×	478	20	×
133	10	÷	306	45 .0	RCL . 0	479	44 8	STO 8
134	44 .4	STO . 4	307	40	+	480	42,21, 7	f LBL 7
135	45,10, 1	RCL ÷ 1	308	22 .3	GTO . 3	481	31	R/S
136	45 .1	RCL . 1	309	43 32	g RTN	482	44 .3	STO . 3
137	45 4	RCL 4	310	42,21, 4	f LBL 4	483	1	1
138	14	yˣ	311	45 .3	RCL . 3	484	45 .3	RCL . 3
139	20	×	312	44 3	STO 3	485	13	10ˣ
140	44 .5	STO . 5	313	45 .5	RCL . 5	486	30	−
141	1	1	314	44 5	STO 5	487	45 4	RCL 4
142	43,30, 8	g TEST x<y	315	45 .4	RCL . 4	488	20	×
143	32 .9	GSB . 9	316	44 4	STO 4	489	1	1
144	1	1	317	45 .6	RCL . 6	490	40	+
145	45 .5	RCL . 5	318	44 6	STO 6	491	43 13	g LOG
146	30	−	319	45 .7	RCL . 7	492	30	−
147	43 12	g LN	320	44 7	STO 7	493	45 6	RCL 6
148	16	CHS	321	32 .1	GSB . 1	494	10	÷
149	45,10, .1	RCL ÷ . 1	322	42,21, .4	f LBL . 4	495	45 1	RCL 1
150	44 5	STO 5	323	31	R/S	496	20	×
151	45,20, .2	RCL × . 2	324	45 .1	RCL . 1	497	44 9	STO 9
152	16	CHS	325	14	yˣ	498	22 7	GTO 7
153	12	eˣ	326	45 .0	RCL . 0	499	43 32	g RTN
154	45,20, .2	RCL × . 2	327	20	×	500	42,21,15	f LBL E
155	45,20, 9	RCL × 9	328	22 .4	GTO . 4	501	36	ENTER
156	45 .4	RCL . 4	329	43 32	g RTN	502	36	ENTER
157	45 .2	RCL . 2	330	42,21, 5	f LBL 5	503	36	ENTER
158	45 4	RCL 4	331	45 0	RCL 0	504	6	6
159	14	yˣ	332	44 3	STO 3	505	30	−
160	20	×	333	45 .5	RCL . 5	506	20	×
161	40	+	334	44 5	STO 5	507	8	8
162	44 .6	STO . 6	335	45 1	RCL 1	508	40	+
163	45 2	RCL 2	336	44 4	STO 4	509	20	×
164	45,10, 3	RCL ÷ 3	337	45 .6	RCL . 6	510	4	4
165	3	3	338	44 6	STO 6	511	20	×
166	0	0	339	45 .8	RCL . 8	512	7	7
167	20	×	340	44 7	STO 7	513	48	.
168	45,10, 6	RCL ÷ 6	341	32 .1	GSB . 1	514	5	5
169	11	√x̅	342	42,21, .5	f LBL . 5	515	30	−
170	45,20, .6	RCL × . 6	343	31	R/S	516	43 32	g RTN
171	45 .6	RCL . 6	344	45 .1	RCL . 1
172	45,30, 2	RCL − 2	345	20	×