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Root-finding algorithm
Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates 1 / x {\textstyle
Fast_inverse_square_root
Algorithms for calculating square roots
may be preferable to compute the inverse square root instead. Other methods are available to compute the square root digit by digit, or using Taylor series
Square_root_algorithms
Number whose square is a given number
square roots in this manner yields the Spiral of Theodorus depicted above. Apotome (mathematics) Cube root Fast inverse square root Functional square
Square_root
1999 video game
original copy of the game to play it as intended. Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF
Quake_III_Arena
Video game engine
calculation of these shaders, id Tech 3 implements a specific fast inverse square root function, which attracted a significant amount of attention in
Id_Tech_3
IEEE standard for floating-point arithmetic
cases of underflow. See Fast inverse square root and Methods of computing square roots#Iterative methods for reciprocal square roots As an implementation
IEEE_754
Code-generating large language model by OpenAI
example the model outputted the training data code implementing the fast inverse square root algorithm, including comments and an incorrect copyright notice
OpenAI_Codex_(language_model)
Matrix with a multiplicative inverse
or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it can be multiplied by its inverse matrix to yield the
Invertible_matrix
Algorithm for finding zeros of functions
Aitken's delta-squared process Bisection method Euler method Fast inverse square root Fisher scoring Gradient descent Integer square root Kantorovich theorem
Newton's_method
32-bit computer number format
and bit-shifting can yield an approximation to reciprocal square root (fast inverse square root), commonly required in computer graphics. ISO/IEC 10967
Single-precision floating-point format
Single-precision_floating-point_format
Arithmetic operation
numbers such as real, complex, or ordinal numbers. The two inverses of tetration are called super-root and super-logarithm. They are respectively analogous
Tetration
Exponent of a power of two
{\displaystyle \sigma \approx 0.043} would halve the maximum error. The fast inverse square root algorithm uses this idea, with a different correction term that
Binary_logarithm
Numeric value with an unclear meaning
redirect targets Enumerated type – Named set of data type values Fast inverse square root – Root-finding algorithm Magic number – Structure of information stored
Magic_number_(programming)
Mathematical concept
function is bijective and so, invertible. The inverse function here is called the (positive) square root function and is denoted by x ↦ x {\displaystyle
Inverse_function
Part of a number in scientific notation
allows for a fast approximation of the base-2 logarithm, leading to algorithms e.g. for computing the fast square-root and fast inverse-square-root. The implicit
Significand
American computer programmer and video game developer (born 1970)
first used in Enemy Territory: Quake Wars. Quake 3 popularized the fast inverse square root algorithm. Carmack's engines have also been licensed for use in
John_Carmack
Discrete Fourier transform algorithm
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform
Fast_Fourier_transform
In computing, a condition where an expression does not have a correct value
undefined behaviour. For example, passing a negative number to the fast inverse square root function will produce a number. Not a very useful number, but the
Undefined_value
Generalisation of Fourier transform to any ring
attributes of the complex DFT, including the inverse transform, the convolution theorem, and most fast Fourier transform (FFT) algorithms, depend only
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
Arithmetic operation, inverse of nth power
number x of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree
Nth_root
Bernoulli's method to find the root of greatest modulus. The inverse power method with shifts, which finds some smallest root first, is what drives the complex
Polynomial_root-finding
Technique circumventing programming language data typing
Practical examples of floating-point punning include fast inverse square root popularized by Quake III, fast FP comparison as integers, and finding neighboring
Type_punning
Inverse functions of sin, cos, tan, etc.
mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of
Inverse trigonometric functions
Inverse_trigonometric_functions
Number with an integer power equal to 1
inverse golden ratio and minus golden ratio. For n = 8, for any root of unity z + z equals to either 0, ±2, or ±√2 (D = 2). For n = 12, for any root of
Root_of_unity
y2)1/2 Alpha max plus beta min algorithm — approximates hypot(x,y) Fast inverse square root — calculates 1 / √x using details of the IEEE floating-point system
List of numerical analysis topics
List_of_numerical_analysis_topics
Compiler optimization
expensive computation (3 ** x) to be replaced by the cheaper (3 * z). Fast inverse square root Hacker's Delight Partial evaluation Steven Muchnick; Muchnick and
Strength_reduction
Arithmetic operation
identity denoted 1 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element x is standardly
Exponentiation
Fixed number that has received a name
encounter during pre-college education in many countries. The square root of 2, often known as root 2 or Pythagoras' constant, and written as √2, is the unique
Mathematical_constant
Non-sinusoidal waveform
than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse). A triangle wave of period p that spans
Triangle_wave
Empirical thermodynamic model of gas-transport phenomenon
experimentally that the rate of effusion of a gas is inversely proportional to the square root of the molar mass of its particles. This formula is stated
Graham's_law
arccos – inverse cosine function. arccosec – inverse cosecant function. (Also written as arccsc.) arccot – inverse cotangent function. arccsc – inverse cosecant
List of mathematical abbreviations
List_of_mathematical_abbreviations
Matrix of partial derivatives of a vector-valued function
the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
{\displaystyle \lambda (n).} If x is a kth root of unity modulo n, then x is a unit (invertible) whose inverse is x k − 1 {\displaystyle x^{k-1}} . That
Root_of_unity_modulo_n
Early scientific pocket calculator
calculators, introduced in 1973, which had featured scientific notation, squares, square root, and reciprocals, but had no trig or log functions, and lacked other
TI_SR-50
preceding digits Square and Nth root of a number: Alpha max plus beta min algorithm: an approximation of the square-root of the sum of two squares Methods of
List_of_algorithms
Function in discrete mathematics
crucially on the availability of a fast algorithm to compute discrete Fourier transforms and their inverses, a fast Fourier transform. When the DFT is
Discrete_Fourier_transform
In mathematics, invariant of square matrices
two square matrices is the product of their determinants. A square matrix is invertible if and only if its determinant has a multiplicative inverse. In
Determinant
Non-sinusoidal waveform
a sawtooth wave ramps upward and then sharply drops. In a reverse (or inverse) sawtooth wave, the wave ramps downward and then sharply rises. It can
Sawtooth_wave
Real square matrix whose columns and rows are orthogonal unit vectors
its transpose is equal to its inverse: Q T = Q − 1 , {\displaystyle Q^{\mathrm {T} }=Q^{-1},} where Q−1 is the inverse of Q. An orthogonal matrix Q is
Orthogonal_matrix
mapping x to f (g(x)). Inverse function: is declared by "doing the reverse" of a given function (e.g. arcsine is the inverse of sine). Implicit function:
List_of_types_of_functions
Laws describing planetary orbits
inverse-square law of gravitation, being a consequence just of the radial nature of that law, whereas the other laws do depend on the inverse-square form
Kepler's laws of planetary motion
Kepler's_laws_of_planetary_motion
Two raised to an integer power
1/8, 1/16, etc. Sometimes these are called inverse powers of two because each is the multiplicative inverse of a positive power of two. Most computers
Power_of_two
probability Inverse probability weighting Inverse relationship Inverse-chi-squared distribution Inverse-gamma distribution Inverse transform sampling Inverse-variance
List_of_statistics_articles
a one-way model involving a square root operator. They are obtained by applying rational approximations to the square root operator. After a one-way model
Beam_propagation_method
Square matrix constructed from a monic polynomial
V {\displaystyle C(p)=V^{-1}\!DV} : they are the column vectors of the inverse Vandermonde matrix V − 1 = [ w 1 T ⋯ w n T ] {\displaystyle V^{-1}=[w_{1}^{T}\cdots
Companion_matrix
Number divisible only by 1 and itself
Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit. Fibonacci took the innovations from Islamic mathematics
Prime_number
Mathematical function, inverse of an exponential function
1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b. The logarithm base 10 is called the decimal
Logarithm
such as peak repetitive off voltage, peak repetitive reverse voltage, root mean square (RMS) on-state current, and temperature junction. Peak repetitive off
Quadrac
Matrix decomposition
so-called rank-revealing QR algorithms. Compared to the direct matrix inverse, inverse solutions using QR decomposition are more numerically stable as evidenced
QR_decomposition
Formula in calculus
c z 2 {\displaystyle {\sqrt {a+bz+cz^{2}}}} as the composite of the square root function and the function a + b z + c z 2 {\displaystyle a+bz+cz^{2}\
Chain_rule
Hypotenuse of right triangle from its sides
quadrature. A scaled version of this operation gives the quadratic mean or root mean square. It is available in many programming libraries as the hypot function
Pythagorean_addition
Branch of mathematics
reconstruction, each image square is reassembled from the preserved approximate Fourier-transformed components, which are then inverse-transformed to produce
Fourier_analysis
Process of a gas escaping through a small hole
pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight.
Effusion
to Fp, the pth root of a polynomial with zero derivative is obtained by the same substitution on x, completed by applying the inverse of the Frobenius
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
Operation common in numerical signal processing
if, for instance, the square root of the result is then evaluated. When implemented inside a microprocessor, an FMA can be faster than a multiply operation
Multiply–accumulate_operation
Multiplication algorithm
product), and compute the inverse transform C {\displaystyle C} of the array C ^ {\displaystyle {\widehat {C}}} , again using the root of unity g {\displaystyle
Schönhage–Strassen_algorithm
1960s American experimental ground-effect vehicle
The Collins X-112 was built to test the concept. The Airfoil Boat was an inverse-delta aircraft, that is, it had a wing which was triangular in plan but
Collins_X-112
Matrix of second derivatives
mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function,
Hessian_matrix
Vector operator in vector calculus
be negative, such as in pseudo-Riemannian spaces. The reason for the square-root is a bit subtle: it effectively avoids double-counting as one goes from
Divergence
Mathematics concept
strings are drawn. The sequence of these numbers grows inversely proportionally to the square root of k. However, some authors write "the Chvátal–Sankoff
Chvátal–Sankoff_constants
Multivariate derivative (mathematics)
contravariant bases respectively, g i j {\displaystyle g^{ij}} is the inverse metric tensor, and the Einstein summation convention implies summation
Gradient
Generalization of the discrete Fourier transform
inequivalent irreducible representations of G {\displaystyle G} . Then the inverse Fourier transform at an element a {\displaystyle a} of G {\displaystyle
Fourier transform on finite groups
Fourier_transform_on_finite_groups
Certain vector fields are the sum of an irrotational and a solenoidal vector field
{\hat {\mathbf {F} }}_{l}(\mathbf {k} )=\mathbf {0} .} Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier
Helmholtz_decomposition
Statistical transform
transform was developed as a more computationally efficient alternative to the inverse transform sampling method. The ziggurat algorithm gives a more efficient
Box–Muller_transform
Four-dimensional number system
The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding
Quaternion
Arithmetical operation
rectangle gives its area (in square meters or square feet). Such a product is the subject of dimensional analysis. The inverse operation of multiplication
Multiplication
Number, approximately 1.618
{5}}}{2}}=-0.618033\dots .} The absolute value of the conjugate root is the multiplicative inverse of the golden ratio, 1 {\displaystyle {1}} . If two quantities
Golden_ratio
In mathematics, a non-algebraic number
numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation
Transcendental_number
Matrix decomposition
minimization is the lowest reliable eigenvalue. In measurement systems, the square root of this reliable eigenvalue is the average noise over the components
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Probabilistic primality test
polynomials). Here follows a more elementary proof. Suppose that x is a square root of 1 modulo n. Then: ( x − 1 ) ( x + 1 ) = x 2 − 1 ≡ 0 ( mod n ) . {\displaystyle
Miller–Rabin_primality_test
Number of partitions of an integer
exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function;
Partition function (number theory)
Partition_function_(number_theory)
Calculus of functions generalization
U} containing f ( a ) {\displaystyle f(a)} . The inverse function theorem then says that the inverse function f − 1 {\displaystyle f^{-1}} is differentiable
Calculus_on_Euclidean_space
Mathematical function, denoted exp(x) or e^x
exp x ⋅ exp y {\displaystyle \exp(x+y)=\exp x\cdot \exp y} . Its inverse function, the natural logarithm, ln {\displaystyle \ln } or log {\displaystyle
Exponential_function
Matrix decomposition method
algorithms, but avoids extracting square roots. For this reason, the LDL decomposition is often called the square-root-free Cholesky decomposition. For
Cholesky_decomposition
Attraction of masses and energy
that is proportional to the product of their masses and inversely proportional to the square of the distance between them. Scientists are looking for
Gravity
2017 research paper by Google
first 4000 (warmup) steps and decreased it proportionally to the inverse square root of the current step number. Dropout layers were applied to the output
Attention_Is_All_You_Need
Low-level shading language
MUL - multiply POW - exponentiate RCP - reciprocal RSQ - reciprocal square root SGE - set on greater than or equal SLT - set on less than SUB - subtract
ARB_assembly_language
Probability distribution
{\displaystyle \sigma ^{2}=\alpha \theta ^{2}=\alpha /\beta ^{2}} The square root of the inverse shape parameter gives the coefficient of variation: σ / μ = α
Gamma_distribution
Signal with equal energy per octave
amplitudes fall off inversely with the square root of frequency u (so that power, which is the square of amplitude, falls off inversely with frequency) and
Pink_noise
Adaptive filter algorithm for digital signal processing
used in real-time applications because of the number of division and square-root operations which comes with a high computational load. The algorithm
Recursive least squares filter
Recursive_least_squares_filter
Platform-independent fixed-point math library
platform-independent fixed-point math library aimed at developers wanting to perform fast non-integer math on platforms lacking a (or with a low performance) FPU.
Libfixmath
Number system extending the rational numbers
{\textstyle p^{n^{2}}} from the inverse modulo p n . {\textstyle p^{n}.} The same method can be used for computing the p-adic square root of an integer that is
P-adic_number
Welding using electromagnetic induction
and therefore heating, penetrates from the surface is inversely proportional to the square root of the frequency. The temperature of the metals being
Induction_welding
Mechanical analog computer
square root, it may be possible to read from a D scale to an R1 scale running from 1 to square root of 10 or to an R2 scale running from square root of
Slide_rule
Algorithm to multiply two numbers
has a (2m)th root of unity. This speeds up computation and reduces the time complexity. However, these latter algorithms are only faster than Schönhage–Strassen
Multiplication_algorithm
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
and logarithmic functions, real and complex multiplication, division, square-root calculation, solution of linear systems, eigenvalue estimation, singular
CORDIC
Smallest amount of fissile material needed to sustain a nuclear reaction
144 kilograms (317 lb), for example. The critical mass is inversely proportional to the square of the density. If the density is 1% more and the mass 2%
Critical_mass
Optimization algorithm
gradient method is typically determined by a square root of the condition number, i.e., is much faster. Both methods can benefit from preconditioning
Gradient_descent
n {\displaystyle a^{2}-n} is a quadratic non-residue, so there is no square root in F p {\displaystyle \mathbf {F} _{p}} . This ω {\displaystyle \omega
Cipolla's_algorithm
Matrix representing a Euclidean rotation
{1-Q_{xx}-Q_{yy}+Q_{zz}}}\right|\end{aligned}}} Alternatively, use a single square root and division t = tr Q = Q x x + Q y y + Q z z r = 1 + t s = 1 2 r w
Rotation_matrix
Speed of sound wave through elastic medium
directions in three dimensions, the intensity drops in proportion to the inverse square of the distance. In the ocean, however, there is a layer called the
Speed_of_sound
then it has a root in that interval (Bolzano's theorem). The image of a continuous function over an interval is itself an interval. . inverse trigonometric
Glossary_of_calculus
American television series
(October 9, 2025). "The Chair Company Is The Perfect Conspiracy Thriller". Inverse. Archived from the original on October 13, 2025. Retrieved October 10,
The_Chair_Company
Mathematical method in electrical engineering
{\displaystyle V} are transformed into the time domain, typically using inverse Fast Fourier transforms. The nonlinear devices are then evaluated using the
Harmonic_balance
Arithmetic operation
numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c / b means a × b = c, as long
Division_(mathematics)
Method for evaluating indefinite integrals
algorithm takes over 100 pages. The Risch–Norman algorithm is a simpler, faster, but less powerful variant that was developed in 1976 by Arthur Norman.
Risch_algorithm
Algorithmic runtime requirements for common math procedures
589M. doi:10.1090/S0025-5718-07-02017-0. Bernstein, D.J. "Faster Algorithms to Find Non-squares Modulo Worst-case Integers". Brent, Richard P.; Zimmermann
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Type of matrix representation
matrix and, therefore, has a unique positive-semidefinite Hermitian square root. If A is invertible, then P is positive-definite, thus also invertible
Polar_decomposition
Rational fractions as sums of simple terms
computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently
Partial fraction decomposition
Partial_fraction_decomposition
Differential equation that is linear with respect to the unknown function
special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric
Linear_differential_equation
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FAST INVERSE-SQUARE-ROOT
FAST INVERSE-SQUARE-ROOT
FAST INVERSE-SQUARE-ROOT
FAST INVERSE-SQUARE-ROOT
FAST INVERSE-SQUARE-ROOT
FAST INVERSE-SQUARE-ROOT
FAST INVERSE-SQUARE-ROOT
FAST INVERSE-SQUARE-ROOT
FAST INVERSE-SQUARE-ROOT
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