Intereting Posts

$p$ is irreducible if and only if the only divisors of $p$ are the associates of $p$ and the unit elements of $R$
Proving $a^ab^bc^c\ge(abc)^{(a+b+c)/3}$ for positive real numbers.
What are the numbers before and after the decimal point referred to in mathematics?
A sufficient condition for a function to be of class $C^2$ in the weak sense.
Good introductory probability book for graduate level?
Evaluation of $\lim\limits_{n\to\infty} (\sqrt{n^2 + n} – \sqrt{n^3 + n^2}) $
Is $f(x,y)=(xy)^{2/3}$ differentiable at $(0,0)$?
The transpose of a linear injection is surjective.
Comaximal ideals
Constructing a circle through a given point, tangent to a given line, and tangent to a given circle
How to evaluate $\lim_{ n\to \infty }\frac{a_n}{2^{n-1}}$, if $a_0=0$ and $a_{n+1}=a_n+\sqrt{a_n^2+1}$?
Books for starting with analysis
Inverse of open affine subscheme is affine
Is $f$ measurable with respect to the completion of the product $\sigma$-algebra $\mathcal{A} \times \mathcal{A}$ on $^2$?
Polynomial map is surjective if it is injective

**How does a technical and theoretical study of a project of implementing an algorithm on a specific hardware architecture?**

Function example:

```
%Expresion 1:
y1 = exp (- (const1 + x) ^ 2 / ((conts2 ^ 2)),
y2 = y1 * const3
```

Where x is the input variable,

y2 is the output and const1, const3 const2 and are constant.

I need to determine the error you get in terms of architecture that decides to develop, for example it suppose that is not the same architecture with 11 bits for the exponent and 52 bits for mantissa.

This is the concept of error I handle:

```
Relative Error = (Real Data - Architecture Data) / (Real Data) * 100
```

I consider as ‘Real Data’ the ouput of my algorithm I get from Matlab (Matlab use double precission float point, IEEE 754, 52 bits to mantisa, 11 bits for exponent, one bit fo sign ) with the expression 1 and I consider as ‘Architecture Data’ the ouput of my algorithm running in an particular architecture (For instance a architecture that uses 12 bits for mantisa, 5 bits for exponent and 1 bit for sign)

- Reasoning the calculation of the Hilbert's distance
- Algorithms for Finding the Prime Factorization of an Integer
- Efficiently partition a set into all possible unique pair combinations
- Algorithm for calculating $A^n$ with as few multiplications as possible
- loop invariant fibonacci
- Horse Race question: how to find the 3 fastest horses?

**EDIT:**

NOTE: The kind of algorithm I am refering to are all those which use matemathical functions that can be descompose in terms of addition, multiplication, subtrations and division.

Thank you!

- How to accurately calculate the error function erf(x) with a computer?
- Master Theorem. How is $n\log n$ polynomially larger than $n^{\log_4 3}$
- Find a conformal map from the disc to the first quadrant.
- Proof that the harmonic series is < $\infty$ for a special set..
- Integral with parameter
- Prove no existing a smooth function satisfying … related to Morse Theory
- Definition of Equivalent Norms
- Is factoring polynomials as hard as factoring integers?
- Prob. 15, Chap. 5 in Baby Rudin: Prove that $M_1^2\leq M_0M_2$, where $M_0$, $M_1$, and $M_2$ are the lubs, resp., of …
- How to count derangements?

You have several issues to consider. First, your output format allows very many fewer output values than the double standard. If your exponent is 2-based, your output will be of the form $\pm m*2^e$ where m has 2048 values available and e has 32 values (maybe they range from -15 to +16). The restricted range of e means that you will underflow/overflow quite often. If you are within range, you will still have an error as large as one part in 4096 assuming your calculation is perfect and the only error is representing the output value.

A second issue is precision of the input and error propagation. If your input values are represented in the same format, they will be in error by up to 1 part in 4096 for the same reason. Then when you calculate $f(x)=c_3 \exp {\frac{-(c_1+x)^2}{c_2}}$ (where I absorbed the squaring of $c_2$ into its definition) the error will propagate. You can assess this by evaluating $\frac{\text{d}\ln f(x)}{\text{d}x}=\frac{\text{d}\ln (c_3 \exp {\frac{-(c_1+x)^2}{c_2}})}{\text{d}x}$ to get the fractional error in $f(x)$ depending on the error in $x$

All this assumes your calculation is perfect and the only problem is the accuracy of representing the data. If not, you also need to assess the error that is inherent in the computation.

I specially recomend you reading What Every Computer Scientist Should Know About Floating-Point Arithmetic by David Goldberg. It should illuminate a lot of issues regarding floating-point computations Errors.

This paper is a tutorial on those aspects of floating-point arithmetic (floating-point hereafter) that have a direct connection to systems building. It consists of three loosely connected parts. The first section, Rounding Error, discusses the implications of using different rounding strategies for the basic operations of addition, subtraction, multiplication and division. It also contains background information on the two methods of measuring rounding error, ulps and relative error. The second part discusses the IEEE floating-point standard, which is becoming rapidly accepted by commercial hardware manufacturers. Included in the IEEE standard is the rounding method for basic operations. The discussion of the standard draws on the material in the section Rounding Error. The third part discusses the connections between floating-point and the design of various aspects of computer systems.

I specially recomend you to design your algorithm according The IEEE 754 rounding methods.

I also recomend you to read the third part because it discussed the implications of floating point arithmetic on the design of computer systems.

- How do you prove $\sum \frac {n}{2^n} = 2$?
- Prove that any set of 2015 numbers has a subset who's sum is divisible by 2015
- Find recursive formula for sequence $a_n = \left(\frac23\right)^n + n$
- Verify the identity: $(1-x^2)\dfrac{\partial^2 \Phi}{\partial x^2}-2x\dfrac{\partial\Phi}{\partial x}+h\dfrac{\partial^2}{\partial h^2}(h\Phi)=0$
- Solving: $3^m-2=n^2$
- Limits and exponents and e exponent form
- Limit of Lebesgue integral in $L_1(,m)$
- Absolute Continuity of Finite Borel Measure Characterized by Orthonormal Basis
- Find the bifurcation points for the following system of ODEs
- Maximum number of equilateral triangles in a circle
- Want help for prove formula by combinatoric argument $1\cdot 2+2\cdot 3+3\cdot 4+\dots +n(n+1)=\frac{n(n+1)(n+2)}{3}$
- What do you call numbers such as $100, 200, 500, 1000, 10000, 50000$ as opposed to $370, 14, 4500, 59000$
- I am going to learn these math topics , please suggest me where to start?
- Zeroes of a Particular Function
- How to show that $(a+b)^p\le 2^p (a^p+b^p)$