Function reference

Equarith 1.0.3 searches over a typed catalog of numerical operations. Every operation has a default complexity cost and a numerical domain. A candidate that produces a non-finite value on a required row is rejected unless the selected operation explicitly defines protected behavior.

The identifiers below are also the canonical names used in parsed formulas. Costs are the v1.0.3 defaults; you can edit them before search.

Recommended default selection

Select Defaults in the function catalog to apply this conservative recommended set:

• add, subtract, and multiply;
• safe_div;
• square and cube;
• sin and cos;
• safe_log and safe_sqrt; and
• abs.

When capabilities are first loaded, the current selection can include every non-history function. Defaults deliberately narrows it to the list above. Keep functions outside this set only when they make sense for the variables, units, and expected relationship.

Arithmetic

• add(a,b), cost 1: addition; operands and result must be finite.
• subtract(a,b), cost 1: subtraction.
• multiply(a,b), cost 1: multiplication; overflow rejects the candidate.
• divide(a,b), cost 2: strict division; |b| must exceed 1e-12.
• safe_div(a,b), cost 3: returns a / b normally and returns a when |b| <= 1e-12.
• aq(a,b), cost 4: analytic quotient a / sqrt(1 + b²), without a finite-input singularity.

Protected division avoids undefined values but changes the mathematical meaning near zero. Inspect whether that branch is physically defensible.

Powers

• square(x), cost 2: x².
• cube(x), cost 3: x³.
• inv(x), cost 2: 1/x; the input must be non-zero.
• pow(x,n), cost 4: real-valued power. Search-generated exponents are non-zero integers constrained to [-3, 3]; the result must be a finite real number.

Trigonometric functions

• sin(x) and cos(x), cost 2 each.
• tan(x), cost 3; candidates producing a non-finite result are rejected.
• asin(x) and acos(x), cost 3 each; x must be in [-1, 1].
• atan(x), cost 3.
• atan2(y,x), cost 3: quadrant-aware angle.
• sinc(x), cost 4: sin(x)/x with sinc(0) = 1.

Arguments use radians. A trigonometric fit can extrapolate periodically far beyond any physical justification.

Hyperbolic functions

• sinh(x), cosh(x), and tanh(x), cost 3 each.
• asinh(x), cost 4.
• acosh(x), cost 4; x >= 1.
• atanh(x), cost 4; -1 < x < 1.

Overflow or a domain violation rejects the candidate.

Exponential, logarithmic, and root functions

• exp(x), cost 3; overflow rejects the candidate.
• expm1(x), cost 3: accurate exp(x) - 1 near zero.
• safe_exp(x), cost 4: evaluates the exponential after clamping x to [-60, 60].
• log(x), cost 3: natural logarithm for x > 0.
• log2(x) and log10(x), cost 4 each, for x > 0.
• log1p(x), cost 3: accurate log(1 + x) for x > -1.
• safe_log(x), cost 4: log(abs(x) + 1e-12).
• sqrt(x), cost 3, for x >= 0.
• cbrt(x), cost 3: real cube root.
• safe_sqrt(x), cost 4: sqrt(abs(x)).

Safe variants produce a finite value over a wider range, but absolute values, clamps, and epsilon terms alter semantics. Prefer strict operations when the domain constraint itself is meaningful and supported by the data.

Shape and aggregation functions

• abs(x), cost 2.
• relu(x), cost 2: max(0,x).
• sigmoid(x), cost 4: stable logistic transition in [0,1].
• softplus(x), cost 4: stable log(1 + exp(x)).
• gaussian(x), cost 4: exp(-x²).
• ceil(x) and floor(x), cost 2 each.
• round(x), cost 3: halfway cases round away from zero.
• sign(x), cost 2.
• min(a,b) and max(a,b), cost 2 each.
• hypot(a,b), cost 3: overflow-resistant sqrt(a² + b²).
• clamp(x,lower,upper), cost 3; the lower bound must not exceed the upper bound.

Discontinuous operations such as rounding, sign, min, max, and clamp can describe regimes, but they can also create unstable boundaries.

Special functions

• fmod(a,b), cost 4: floating-point remainder; b must be non-zero.
• tgamma(x), cost 4: gamma function; poles and overflow are invalid.
• lgamma(x), cost 4: logarithm of the absolute gamma value; poles are invalid.
• erf(x), cost 4: error function.
• erfc(x), cost 4: complementary error function with stable Gaussian-tail behavior.

Logical and conditional functions

These operations return numerical values and can build piecewise formulas:

• smaller(a,b) and greater(a,b), cost 4: return 0 or 1.
• equal(a,b) and different(a,b), cost 4: use exact floating-point equality and return 0 or 1.
• logical_or(a,b) and logical_and(a,b), cost 4: zero is false and non-zero is true.
• if_else(condition,when_true,when_false), cost 5: selects the second operand when the condition is non-zero, otherwise the third.

Exact equality is rarely robust for measured continuous values. A threshold comparison is usually more meaningful.

History functions

History functions are disabled by default and use original source-row chronology, even when training rows are randomly split or progressively sampled. rolling_std and rolling_slope have cost 5; every other history function below has cost 4.

• delay(expr,N): the expression value N original rows before the current row.
• moving_average(expr,N): mean over the preceding N rows, excluding the current row.
• difference(expr,N): current value minus its value N rows earlier.
• rolling_std(expr,N): population standard deviation over the preceding N rows.
• rolling_min(expr,N) and rolling_max(expr,N): extreme over the preceding N rows.
• rolling_slope(expr,N): least-squares slope over the preceding N rows; N must be at least 2.
• decay(expr,r): exponentially weighted sum of the current and preceding values using powers of r, truncated at Maximum history size.
• ema(expr,r): normalized exponentially weighted average of the current and preceding values.

Window lengths are integers from 1 up to Maximum history size, which can be configured from 1 to 1,000. Decay rates satisfy 0 < r < 1.

The initial warm-up rows and any row whose required history is invalid are excluded before the train/test split. Increasing Maximum history size can therefore reduce the eligible dataset substantially.

Allow target variable in history functions enables autoregressive formulas. Such a formula can be valid for one-step prediction when previous observed targets exist, but invalid for a deployment that does not have them.

History expressions are supported for dataset evaluation, projects, and compatible checkpoints. They are not available for unordered custom prediction points and cannot be exported as standalone executable source in v1.0.3. Presentation exports such as plain formula text and LaTeX remain available.

Complexity costs

Costs express an interpretability preference, not runtime cost alone. The complexity of each formula contributes directly to Pareto construction. Raising the cost of a function makes formulas using it compete at a higher complexity; lowering it makes that operation easier to retain.

Use one coherent cost scheme throughout an experiment. Changing costs can reorder the Pareto front and makes an existing checkpoint incompatible.

For the surrounding controls, read Configuring a search. For generated-code restrictions, read Exporting data and formulas.