A clear path through
the mathematics.

Interpret a signed integer as a position relative to zero.

Add two signed integers.

Subtract one signed integer from another.

Multiply two signed integers.

Divide signed integers when the quotient is an integer and the divisor is nonzero.

Compute a real number’s distance from zero.

Interpret a fraction as a quotient with a nonzero denominator.

Generate an equivalent fraction by scaling its numerator and denominator by the same nonzero integer.

Add two fractions using a common denominator.

Subtract two fractions using a common denominator.

Multiply two fractions.

Divide a fraction by a nonzero fraction.

Evaluate a positive whole-number power.

Evaluate a nonzero base raised to the zero power.

Evaluate a nonzero base raised to a negative integer power.

Rewrite a product of powers with the same nonzero base as one power.

Rewrite a quotient of powers with the same nonzero base as one power.

Rewrite a power raised to an integer power as one power for a nonzero base.

Evaluate a real square root with its domain restriction.

Isolate one scalar variable in a linear equation.

Expand a scalar product over a sum.

Factor a common scalar factor from a sum.

Solve a one-step scalar inequality.

Locate a point from an ordered coordinate pair.

Evaluate a scalar function at a supplied input.

Determine which real inputs a scalar expression allows.

Read the output of a scalar function from its graph.

Compute the slope between two distinct points.

Evaluate a scalar affine rule from its slope and intercept.

Evaluate a composition in the correct order.

Recover an input using an invertible scalar function.

Predict multiplicative change from an exponential rule.

Evaluate a logarithm as an inverse exponent.

Rewrite a positive product using logarithms.

Decide whether an object belongs to a specified finite set.

Form the union of two finite sets.

Form the intersection of two finite sets.

Expand and evaluate a finite indexed sum.

Expand and evaluate a finite indexed product.

Distinguish a universal claim from an existence claim.

Refute a universal claim with a valid counterexample.

Interpret the entries of a vector in a named coordinate system.

Add two vectors of the same dimension.

Scale each coordinate of a vector by a scalar.

Compute the displacement between two coordinate vectors.

Construct a weighted sum of supplied vectors.

Compute the dot product of two equal-length vectors.

Compute the Euclidean length of a vector.

Normalize a nonzero vector to unit length.

Compute the Euclidean distance between two vectors.

Decide whether two vectors are orthogonal using their dot product.

Compute directional similarity for two nonzero vectors.

Interpret a supplied embedding ranking within its stated representation.

Compute the L1 norm of a vector.

Identify the row and column dimensions of a matrix.

Retrieve a matrix entry using its row and column indices.

Add matrices of equal shape.

Multiply each matrix entry by a scalar.

Compute a matrix-vector product from row dot products.

Express a matrix-vector product as a combination of its columns.

Compute the entries of a compatible matrix product.

Transpose a rectangular matrix.

Interpret the coordinate scaling performed by a diagonal matrix.

Construct an outer product from two vectors.

Encode a finite linear system as Ax=b.

Apply one solution-preserving elementary row operation.

Eliminate a variable using elementary row operations.

Identify pivot positions in row-echelon form.

Solve a triangular system by back substitution.

Reduce an echelon matrix to reduced row-echelon form.

Detect a contradictory row in an augmented system.

Parameterize the free variables of a consistent reduced system.

Determine whether a consistent system has a unique solution.

Decide whether a vector lies in the span of supplied vectors.

Check the defining closure conditions for a candidate real subspace.

Test whether a supplied vector list is linearly independent.

Verify a basis for a specified finite-dimensional subspace.

Determine subspace dimension from a basis.

Find a basis for the column space of a matrix.

Determine matrix rank from its pivots.

Check additivity and homogeneity for a candidate map.

Represent a composition of linear maps as an ordered matrix product.

Separate the linear part from the translation in an affine map.

Compute the orthogonal projection onto a nonzero vector's span.

Verify that a supplied basis is orthonormal.

Project a vector using a supplied orthonormal basis.

Identify the closest attainable output in an inconsistent linear system.

Use supplied reduced QR factors to solve a full-column-rank least-squares problem.

Form the normal equations for a linear least-squares problem.

Interpret the three transformations in a supplied SVD.

Infer rank from the nonzero singular values of a matrix.

Construct a rank-k approximation from supplied singular components.

Compute a matrix's Frobenius norm.

Compute truncated-SVD Frobenius error from discarded singular values.

Interpret the ratio of largest to smallest nonzero singular values for a full-rank square system.

Compute an average rate of change over an interval.

Infer a finite limit from nearby values while distinguishing the endpoint value.

Determine a one-sided limit from a piecewise graph.

Evaluate a finite limit using valid algebraic limit laws.

Form the difference quotient for a scalar function.

Compute a simple derivative from a difference-quotient limit.

Diagnose failure of differentiability from unequal one-sided slopes.

Differentiate a constant scalar function.

Differentiate a scalar power function on its real domain.

Differentiate a sum of differentiable scalar functions.

Differentiate a constant multiple of a differentiable function.

Differentiate a product of two scalar functions.

Differentiate a composition of two scalar functions.

Compute the second derivative of a scalar function.

Evaluate a scalar function at a vector input.

Identify inputs sharing the same scalar output.

Compute a partial derivative with other coordinates held fixed.

Assemble a gradient from partial derivatives in coordinate order.

Compute a directional rate using a unit direction.

Construct the Jacobian of a vector-valued map.

Compute the derivative of a composed vector map using Jacobians.

Represent a scalar expression as a directed graph of elementary operations.

Evaluate a small computational graph in dependency order.

Compute one graph operation's local derivative with respect to an input.

Propagate a scalar-output sensitivity backward through a chain.

Accumulate sensitivities at an input used by multiple graph branches.

Propagate a supplied output sensitivity through a vector operation.

Estimate a derivative with a centered finite difference.

Diagnose a mismatch between analytic and numerical derivatives using a stated tolerance.

Specify outcomes fine enough to describe a finite experiment.

Represent an event as a subset of a sample space.

Check a finite probability assignment for nonnegativity and unit total.

Compute an event probability by summing disjoint outcome masses.

Compute a two-event union probability with overlap correction.

Determine whether two events are mutually exclusive.

Compute an event probability within a positive-probability conditioning event.

Compute an intersection using a conditional probability.

Verify that a collection of events partitions the sample space.

Compute an event probability across an exhaustive disjoint case split.

Reverse a conditional probability using a prior and total evidence probability.

Explain a posterior result using the prevalence of the hypothesized event.

Check independence using the product condition.

Check independence within a stated conditioning event.

Define a numerical random variable on a finite sample space.

Construct a discrete random variable's probability mass function.

Compute a discrete cumulative distribution at a threshold.

Compute P(a < X ≤ b) by subtracting supplied CDF values at the endpoints.

Compute a finite random variable's expectation.

Compute the expectation of a function of a discrete random variable.

Compute the expectation of a linear combination without assuming independence.

Compute variance as expected squared deviation from the mean.

Express spread in the original variable's units.

Compute variance after a scalar shift and scale.

Interpret a supplied area under a continuous density curve as an interval probability.

Identify a normal model's location and scale parameters.

Convert a normal threshold to standard-normal units.

Compute a normal interval probability from a supplied standard-normal CDF.

Construct a joint discrete distribution over paired values.

Marginalize a joint mass table over one variable.

Check factorization of a discrete joint distribution into its marginals.

Identify a chi-squared variable as a sum of squared independent standard-normal variables.

Interpret a supplied standard-normal to scaled-chi-squared ratio as a t variable under independence.

Compute an expectation of a function using a finite joint table.

Compute covariance from a joint distribution.

Compute variance of a sum with a supplied covariance.

Assemble the covariance matrix of a finite random vector.

Check the identical-distribution and independence assumptions in a sampling story.

Compute the variance of an IID sample mean with finite variance.

Bound a nonnegative variable's upper tail from its mean.

Bound a deviation probability using finite variance.

Interpret convergence of an IID sample average under a stated finite-variance assumption.

Form a normal approximation for a standardized IID sum with finite positive variance.

Diagnose a normal approximation undermined by the sampling assumptions or heavy tails.

Estimate an expectation with a supplied set of independent simulated draws.

Compute the information associated with a positive-probability event.

Compute entropy of a finite probability distribution.

Compute expected negative log probability under a supplied target distribution.

Compute a finite-distribution KL divergence with valid support handling.

Classify an observed variable by its numerical or categorical meaning.

Identify the target population and observed sample in a study.

Compute a sample arithmetic mean.

Compute a sample median using the stated even-size convention.

Locate a sample quantile under a supplied convention.

Compute sample variance using an explicitly stated denominator.

Distinguish a data-dependent estimator from its realized estimate.

Identify the distribution induced by repeatedly sampling an estimator.

Evaluate candidate parameters using a fixed discrete sample's probability model.

Compare candidate parameters using a fixed continuous sample density.

Express an independent-sample likelihood as a sum of log terms.

Select a posterior mode from a supplied likelihood and prior.

Normalize posterior weights over a finite parameter grid.

Interpret the standard deviation of an estimator's sampling distribution.

Interpret confidence through repeated coverage of a fixed parameter.

Construct a one-sample t interval under stated sampling conditions.

Construct a bootstrap sampling approximation for an IID sample statistic.

Read a percentile interval from supplied bootstrap replicates with stated limitations.

Specify a null and alternative for a stated population claim.

Compute a supplied test statistic measuring discrepancy from a null.

Compute a tail probability under a specified null distribution.

Interpret the magnitude of an effect in meaningful units.

Apply a Bonferroni familywise-error bound to a stated test family.

Specify a conditional mean model separating predictors from residual noise.

Encode supplied predictors and an intercept as a design matrix.

Obtain coefficients for a full-rank regression using a supplied QR least-squares procedure.

Compute residuals from fitted predictions and observations.

Diagnose coefficient instability caused by dependent or nearly dependent predictors.

Separate model fitting from an independent evaluation set.

Aggregate validation error from a supplied fold scheme without test-set reuse.

Identify preprocessing or selection that uses unavailable evaluation information.

Convert a linear score to a probability through a supplied logistic link.

Compute binary negative log-likelihood for logistic predictions.

Assess whether predicted probabilities match observed frequencies in supplied bins.

Center each data feature using training-sample means.

Find principal directions from a supplied SVD of centered data.

Project centered observations into a chosen principal-component basis.

Compute a retained-variance fraction from supplied singular values.

Express a scalar objective for specified decision variables.

Identify decisions satisfying all stated constraints.

Distinguish the best objective value from the decision attaining it.

Distinguish neighborhood optimality from optimality over the feasible set.

Form an average per-example loss over a fixed dataset.

Check whether a line segment between feasible points remains feasible.

Check the convexity inequality for a supplied scalar example.

Apply the zero-gradient necessary condition to a differentiable interior local optimum.

Check whether a direction has a negative first-order objective change.

Compute one gradient descent update.

Diagnose overshooting from a step size on a one-dimensional quadratic.

Assess a proposed stopping decision using gradient size and supplied tolerances.

Compute a momentum update using a stated velocity convention.

Compute a single-example gradient for an empirical objective.

Average a minibatch's per-example gradients.

Interpret batch-size effects under a stated independent-gradient variance model.

Compute a parameter update using a supplied sampled gradient.

Compute an update with supplied per-coordinate second-moment scaling.

Compute an Adam parameter update from supplied bias-corrected moment estimates.

Correct a zero-initialized exponential moving average using a supplied formula.

Rescale a gradient to satisfy a supplied norm threshold.

Compute the gradient contribution of a stated quadratic penalty.

Solve a small penalized least-squares system with a stated treatment of the intercept.

Compute an L1-penalized objective for supplied coefficients.

Identify valid subgradients of absolute value, including at zero.

Apply a supplied soft-thresholding operator to one scalar coordinate.

Compute a supplied proximal-gradient step for an L1-regularized objective.