Mathematics · Probability

Kullback–Leibler Divergence from Cross-Entropy Gap cross entropy H(P,Q) Solver

Rearrange the kullback–leibler divergence from cross-entropy gap relationship and solve for cross entropy h(p,q).

Runs locally
Your numbers

Inputs and results stay in this browser. Change one value at a time to explore the relationship.

Your inputCalculatedPassed forward in chains
cross entropy H(P,Q)2.8
Reconstructed KL divergence D(P||Q)0.6

Calculation steps

  1. Use a=c+b with KL divergence D(P||Q)=0.5999999999999996 and source entropy H(P)=2.2.
  2. cross entropy H(P,Q)=2.8.
  3. Substitution into c=a−b reconstructs 0.5999999999999996.

Understand Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q)

One idea, three depths

Choose how deeply to explain Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q)

Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q): Rearrange the kullback–leibler divergence from cross-entropy gap relationship and solve for cross entropy h(p,q).

Age 5Explain it to a 5-year-oldStart with a picture

Imagine using Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q) to answer this question: rearrange the kullback–leibler divergence from cross-entropy gap relationship and solve for cross entropy h(p,q)? Enter KL divergence D(P||Q) and source entropy H(P); the calculator shows cross entropy H(P,Q). For example: cross entropy H(P,Q)=2.8 and source entropy H(P)=2.2 produce KL divergence D(P||Q)=0.5999999999999996. The answer tells you cross entropy H(P,Q).

Age 15Explain it to a 15-year-oldConnect it to the formula

Kullback–Leibler divergence equals cross entropy minus source entropy when the same logarithm base is used. This page isolates cross entropy h(p,q) and verifies it in the original relationship. The rule is a=c+b. Its input values are KL divergence D(P||Q), source entropy H(P), and the main result is cross entropy H(P,Q). For example: cross entropy H(P,Q)=2.8 and source entropy H(P)=2.2 produce KL divergence D(P||Q)=0.5999999999999996.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated kullback–leibler divergence from cross-entropy gap: solve cross entropy h(p,q) relation over the valid real-number domain stated below. The implemented relation is a=c+b, evaluated from KL divergence D(P||Q), source entropy H(P) to produce cross entropy H(P,Q). Kullback–Leibler divergence equals cross entropy minus source entropy when the same logarithm base is used. This page isolates cross entropy h(p,q) and verifies it in the original relationship. The order of P and Q matters, and zero model probability on positive source mass makes divergence infinite.

Inputs and valid domain

  • KL divergence D(P||Q) must be a finite real number.
  • source entropy H(P) must be a finite real number.

Important boundary: The order of P and Q matters, and zero model probability on positive source mass makes divergence infinite.

The formula

a=c+b

How the calculator works through it

It substitutes KL divergence D(P||Q), source entropy H(P) into the formula and exposes every numerical step above. The main output is cross entropy H(P,Q), accompanied by Reconstructed KL divergence D(P||Q).

Read the result correctly

The cross entropy H(P,Q) is the direct answer to “rearrange the kullback–leibler divergence from cross-entropy gap relationship and solve for cross entropy h(p,q).” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.

A worked check

cross entropy H(P,Q)=2.8 and source entropy H(P)=2.2 produce KL divergence D(P||Q)=0.5999999999999996.

Where this model stops being reliable

The order of P and Q matters, and zero model probability on positive source mass makes divergence infinite.

Learn it by changing one value

Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.

Dictionary terms behind this calculator

Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.

These foundations help you understand why Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q) works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

  • Reading formulas and substituting values

    Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q) uses a=c+b. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.

    Review this foundation about 4 min

Strong support

  • Probability as a modelled proportion

    Probability rules are needed to interpret what the Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q) result says about possible outcomes.

    Review this foundation about 5 min

Optional enrichment

  • Ordered arrangements

    Counting ordered arrangements can extend Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q) to more detailed sample spaces and event models.

    Review this foundation about 5 min
Learn the missing foundationsI already know these — show the code

Mathematics → algorithm → program

Implement this calculation in code

These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.

Algorithm

  1. Read KL divergence D(P||Q), source entropy H(P).
  2. Evaluate the principal relationship: a=c+b.
  3. Return cross entropy H(P,Q) and check the domain conditions described above.
Python
            from math import *

def kl_cross_entropy_gap_solve_a(c, b) -> float:
    return (c + b)

assert abs(kl_cross_entropy_gap_solve_a(0.5999999999999996, 2.2) - 2.8) < 1e-6 * max(1.0, abs(2.8))
          
Current calculator valuesUpdates when you change an input above.
              
            
C
            #include <assert.h>
#include <math.h>

double kl_cross_entropy_gap_solve_a(double c, double b) {
    return (c + b);
}

int main(void) {
    const double expected = 2.8;
    const double actual = kl_cross_entropy_gap_solve_a(0.5999999999999996, 2.2);
    assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
          
Current calculator valuesUpdates when you change an input above.
              
            
C++
            #include <cassert>
#include <cmath>
#include <numbers>

double kl_cross_entropy_gap_solve_a(double c, double b) {
    return (c + b);
}

int main() {
    constexpr double expected = 2.8;
    const double actual = kl_cross_entropy_gap_solve_a(0.5999999999999996, 2.2);
    assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
          
Current calculator valuesUpdates when you change an input above.
              
            
Linux x86-64 assembly

x86-64 NASM · System V ABI · Linux · SSE2 with libm where required

            ; double kl_cross_entropy_gap_solve_a(double c, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global kl_cross_entropy_gap_solve_a
section .text

kl_cross_entropy_gap_solve_a:
    push rbp
    mov rbp, rsp
    sub rsp, 32
    movsd [rbp-8], xmm0
    movsd [rbp-16], xmm1
    movsd xmm0, [rbp-8]
    addsd xmm0, [rbp-16]
    movsd [rbp-24], xmm0
    movsd xmm0, [rbp-24]
    leave
    ret
          
Current calculator valuesUpdates when you change an input above.
              
            
MATLAB
            function result = kl_cross_entropy_gap_solve_a(c, b)
    result = (c + b);
end
          
Current calculator valuesUpdates when you change an input above.
              
            
Wolfram Language
            ClearAll[mwCalculate];
mwCalculate[c_, b_] := (c + b);
          
Current calculator valuesUpdates when you change an input above.
              
            

Continue in mathematical software

The downloaded file includes your current inputs and first calculated result. It is created locally.

Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.

Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations

Standards, reading and academic references

Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.

Introductory Statistics 2e

Read the free OpenStax statistics textbook
Cite this book
APA 7
Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
MLA 9
Illowsky, Barbara, and Susan Dean. Introductory Statistics 2e. OpenStax, 2023, https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
Chicago author-date
Illowsky, Barbara, and Susan Dean. 2023. Introductory Statistics 2e. Houston, TX: OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.

OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.

Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS

These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.

APA 7

MW SysArc. (2026, July 21). Kullback–Leibler Divergence from Cross-Entropy Gap cross entropy H(P,Q) Solver. MW SysArc Tools. https://math.mwsysarc.com/probability/kl-cross-entropy-gap-cross-entropy-h-p-q-solver

MLA 9

MW SysArc. “Kullback–Leibler Divergence from Cross-Entropy Gap cross entropy H(P,Q) Solver.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/probability/kl-cross-entropy-gap-cross-entropy-h-p-q-solver. Accessed 31 Aug. 2026.

Chicago 17

MW SysArc. “Kullback–Leibler Divergence from Cross-Entropy Gap cross entropy H(P,Q) Solver.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/probability/kl-cross-entropy-gap-cross-entropy-h-p-q-solver.

Harvard

MW SysArc (2026) ‘Kullback–Leibler Divergence from Cross-Entropy Gap cross entropy H(P,Q) Solver’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/probability/kl-cross-entropy-gap-cross-entropy-h-p-q-solver (Accessed: 31 August 2026).

BibTeX and RIS records

BibTeX

@misc{mwsysarc_kl_cross_entropy_gap_solve_a_2026,
  author = {{MW SysArc}},
  title = {Kullback–Leibler Divergence from Cross-Entropy Gap cross entropy H(P,Q) Solver},
  howpublished = {MW SysArc Tools},
  year = {2026},
  url = {https://math.mwsysarc.com/probability/kl-cross-entropy-gap-cross-entropy-h-p-q-solver},
  note = {Published July 21, 2026; accessed August 31, 2026}
}

RIS

TY  - ELEC
AU  - MW SysArc
TI  - Kullback–Leibler Divergence from Cross-Entropy Gap cross entropy H(P,Q) Solver
T2  - MW SysArc Tools
PY  - 2026
DA  - 2026-07-21
Y2  - 2026-08-31
UR  - https://math.mwsysarc.com/probability/kl-cross-entropy-gap-cross-entropy-h-p-q-solver
N1  - Published July 21, 2026
ER  -

Clear answers

Frequently asked questions

What does the Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q) do?

Rearrange the kullback–leibler divergence from cross-entropy gap relationship and solve for cross entropy h(p,q).

How does the Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q) work?

The calculator applies a=c+b. Kullback–Leibler divergence equals cross entropy minus source entropy when the same logarithm base is used. This page isolates cross entropy h(p,q) and verifies it in the original relationship.

What can I learn from the Kullback–Leibler Divergence from Cross-Entropy Gap: solve cross entropy H(P,Q)?

It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.

Does MW SysArc receive or store what I enter?

No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.

How should I use the result?

Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.

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