RPN: Absolute Value Equations with the HP 11C and DM41X

RPN: Absolute Value Equations with the HP 11C and DM41X

Introduction: Solving |z * w + y| = x

Today’s blog focuses on solving the absolute value equation:

|z * w + y| = x

for the variable w, and the values of z, y, and x are given and are on the Classic RPN stack.

For example, in the problem |5 * w + 7| = 2, the stack would be set up as:

t: (anything, it doesn’t matter)

z: 5

y: 7

x: 2

(This is why I set the variable as w instead of the usual x.)

One approach is to use memory registers and other uses only stack operations. Today’s algorithm focuses on the latter.

Caution: In the above equation, x will always be non-negative. The equation will never be valid if x is negative.

The Algebra

Solve for w:

|z * w + y| = x

This leads us to solve the two equations:

(I)

z * w + y = -x

z * w = -x – y

w = -x/z – y/z

Let w- = -x/z – y/z = (-x/z) + (-y/z)

(II)

z * w + y = x

z * w = x – y

w = x/z - y/z

Let w+ = x/z – y/z = (x/z) + (-y/z)

Then:

w- = -x/z – y/z

w- = (-x/z) + (-y/z)

w- = (-x/z) + (-y/z) + (-x/z) + (x/z)

w- = 2 * (-x/z) + (x/z) + (-y/z)

w- = 2 * (-x/z) + w+

RPN Code: HP 11C (adoptable for other RPN calculators)

LBL A	001	42, 21, 11	Program start
R↓	002	33
R↓	003	33
X<>Y	004	34
R↓	005	33
1/x	006	15
×	007	20
LST x	008	43, 36
X<>Y	009	34
R↓	010	33
×	011	20
LST x	012	43, 36
R↑	013	43, 33
X<>Y	014	34
R↓	015	33
X<>Y	016	34
CHS	017	16
X<>Y	018	34
+	019	40
ENTER	020	36
ENTER	021	36
LST x	022	43, 36
-	023	30
LST x	024	43, 36
-	025	30
RTN	026	43, 32	Program end

RPN Code: DM41X (HP 41C series, no module is required)

LBL “ASBEQ”

RDN

RDN

X<>Y

RDN

1/x

×

LAST X

X<>Y

RDN

×

LAST X

R↑

X<>Y

RDN

X<>Y

CHS

X<>Y

+

ENTER

ENTER

LAST X

-

LAST X

-

RTN

Notes:

* RDN is shown as R↓

* To enter R↑, press XEQ, ALPHA, R, SHIFT, ENTER, ALPHA

Subroutines Used

We used several techniques to manipulate the stack. They are presented below:

Rotate stack from X, Y, Z, T to Z, X, Y, T

R↓

R↓

X<>Y

R↓

Source:

Ball, John A. Algorithms for RPN Calculators John Wiley & Sons: New York. 1978. ISBN 0-471-03070-8. pg. 78

Multiply Y and Z by 1/X. Stack: X, Y, Z, T → X, Y/X, Z/X, T

1/x

×

LAST X

X<>Y

R↓

×

LAST X

R↑

X<>Y

Change X, Y to Y+ X, Y – X

+

ENTER

ENTER

LAST X

-

LAST X

-

The resulting stack is:

z

z

y + x

y – x

Doing LAST X, -, twice is effectively subtracting whatever is in L register twice.

Examples

|4 * w – 6| = 5

Stack:

z: 4

y: -6

x: 5

Results:

y: 2.75

x: 0.25

|2 * w + 5| = 3

Stack:

z: 2

y: 5

x: 3

Results:

y: -1

x: -4

Eddie

All original content copyright, © 2011-2026. 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.

RPN: HP 11C: Transferring Between Bases (Common/Natural)

RPN: HP 11C: Transferring Between Bases (Common/Natural)

All algorithms were tested with the HP 11C.

Between the Exponential Function and Common-Antilog

e^α = 10^ß

Given α, what is ß?

ß = log(e^α)

RPN:

<input α>

e^x

LOG

Examples (Fix 6):

e^1.05 = 10^ß

ß ≈ 0.456009

e^(-2.2) = 10^ß

ß ≈ -0.955448

Given ß, what is α?

α = ln(10^ß)

RPN:

<input ß>

10^x

LN

Examples (Fix 6):

e^α = 10^5.4

α ≈ 12.433960

e^α = 10^0.366

α ≈ 0.842746

Between the Natural Logarithm and Common Logarithm

log α = ln ß

Given α, what is ß?

ẞ = exp(log α)

RPN:

<input α>

LOG

e^x

Examples (Fix 6):

log 17 = ln ß

ß ≈ 3.422766

log 317 = ln ß

ß ≈ 12.195405

Given ß, what is α?

α = 10^(ln ß)

RPN:

<input ß>

LN

10^x

Examples (Fix 6):

log α = ln 425

α ≈ 1,127,428.915

log α = ln 9.81

α ≈ 192.044677

Eddie

All original content copyright, © 2011-2026. 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.

RPN: HP 11C: Surface Gravity and Escape Velocity

RPN: HP 11C: Surface Gravity and Escape Velocity

EQUATIONS

The surface gravity constant of a celestial object (planet, dwarf planet, star, etc.):

g_p = G * M ÷ R²

The escape velocity of a celestial object:

v_esc = √(2 * G * M ÷ R)

where (using SI units):

g_p: surface gravity (m/s)

v_esc: escape velocity (m/s)

M: (measured) mass of the object (kg)

R: (average) radius of the object (m)

G: Universal Gravitational Constant (G ≈ 6.6743 * 10^-11 N m²/kg² (or m³/(s² kg))

The value of G is the 2022 CODATA value (https://physics.nist.gov/cgi-bin/cuu/Value?bg)

Determining Surface Gravity and Escape Velocity

DERIVATION - Determine the surface gravity constant in terms of escape velocity.

Start with the escape velocity:

v_esc = √(2 * G * M ÷ R)

(v_esc)² = 2 * G * M ÷ R

dividing both sides by 2 (we'll see why this important in a bit):

(v_esc)² ÷ 2 = G * M ÷ R

(v_esc)² * 1/2 = G * M * 1/R

Then insert the square of escape velocity in the equation for the surface velocity:

g_p = G * M ÷ R²

g_p = G * M * 1/R²

g_p = G * M * 1/R * 1/R

g_p = (v_esc)² * 1/2 * 1/R

g_p = (v_esc)² ÷ (2 * R)

The equations will the be:

v_esc = √(2 * G * M ÷ R)

g_p = (v_esc)² ÷ (2 * R)

Set the stack up as:

Y: M (mass, kg)

X: R (radius, m)

The results are shown in the stack:

Y: g_p (surface gravity, m/s²)

X: v_esc (escape velocity, m/s)

Algorithm (done with an HP 11C):

ENTER

ENTER

R↑

2

×

6.6743e-11 (Keys: 6 . 6 7 4 3 EEX 1 1 CHS)

×

R↑

÷

ENTER

√ (view escape velocity)

R↓

x<>y

÷

2

÷ (view surface gravity)

R↑ (set surface gravity in the Y stack, escape velocity in the X stack)

Example:

Estimate the surface gravity constant and escape velocity of Venus.

Venus

Mass ≈ 4.8675 * 10^24 kg

Radius ≈ 6.0518 * 10^6 m

Surface gravity ≈ 8.8704 m/s

Escape velocity ≈ 10361.6414 m/s

Determining a Planet's Radius and Escape Velocity

Problem: Given Earth's surface gravity is defined as 9.80665 m/s and mass of 5.972168 * 10^24. Estimate the radius and escape velocity.

Here we are given g_p and M, and we are tasked with finding R and v_esc.

Start by solving for R:

g_p = G * M ÷ R²

Multiply by R² and divide by g_p. Keep this in mind.

R² = G * M ÷ g_p

Take the square root and solve for the radius.

R = √(G * M ÷ g_p)

Note that:

R² = G * M ÷ g_p

g_p * R² = G * M

2 * g_p * R² = 2 * G * M

2 * g_p * R = 2 * G * M ÷ R

This makes for an easy substitution for v_esc.

v_esc = √(2 * G * M ÷ R) = √(2 * g_p * R)

The equations used are:

R = √(G * M ÷ g_p)

v_esc = √(2 * g_p * R)

The algorithm uses one memory register, I just picked R0 (done with an HP 11C):

STO 0

÷

6.6743e-11 (Keys: 6 . 6 7 4 3 EEX 1 1 CHS)

×

√ (view R)

ENTER

RCL 0

×

2

×

√ (view v_esc)

Set the stack up as:

Y: M (mass, kg)

X: g_p (surface gravity, m/s²)

The results are shown in the stack:

Y: R (radius, m)

X: v_esc (escape velocity, m/s)

Results:

Inputs:

Mass of Earth ≈ 5.972168 * 10^24 kg (enter as the y stack)

Surface Gravity = 9.80665 m/s² (enter as a x stack, and yes, surface gravity of Earth is defined to be exactly 9.80665 m/s²)

Outputs:

Y: Radius of Earth ≈ 6375416.060 m

X: Escape Velocity ≈ 11182.2604 m/s

Sources

The NIST Reference on Constants, Units, and Uncertainty. "Newtonian constant of gravitation" Fundamental Physical Constants. Last updated May 9, 2024. https://physics.nist.gov/cgi-bin/cuu/Value?bg Retrieved September 4, 2025.

Research & Education Association. The Essentials of Astronomy Piscataway, New Jersey. 2004. ISBN 0-87891-965-1

Eddie

All original content copyright, © 2011-2026. 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.