@/math: numeric functions and constants

The math module is ambient: its methods and constants are available without an import. Floating-point operations use IEEE-754 double-precision semantics, including JavaScript's NaN and positive or negative infinity results for out-of-domain and overflow cases.

Float methods

float provides the trigonometric methods sin, cos, tan, asin, acos, and atan; the hyperbolic methods sinh, cosh, tanh, asinh, acosh, and atanh; and floor, ceil, and round. exp() computes (e^x), while ln() computes the natural logarithm. log(base) is defined in Nymph as:

func log(base: float): float = this.ln() / base.ln()

atan2(y, x) is a top-level function whose operands are interpreted in that order. Integer trigonometric and hyperbolic convenience methods convert their receiver to float and return a float.

The host implementations receive canonical boxed Nymph values and return canonical float or int boxes. Raw JavaScript numbers exist only while a host math primitive is being called; source operands are evaluated once in source order.

Constants

• pi, tau, e, and phi are the usual mathematical constants.
• max_float is the largest finite positive float (Number.MAX_VALUE).
• min_float is the most negative finite float (-Number.MAX_VALUE).
• min_positive_float is the smallest positive representable float (Number.MIN_VALUE, including subnormal values).
• max_int and min_int are the language's nominal signed integer bounds.

The external float constants are immutable boxed values initialized once per generated program and shared by every reference.

Complex numbers and powers

import std/math/complex with (Complex) provides the canonical Complex.new(real: float, imaginary: float) constructs a complex number. It supports addition, subtraction, multiplication, division, negation, conjugation, magnitude, and the exact scalar exponentiation matrix documented under Operators. The same import supplies the real base/float exponent rows because those rows return Complex.

Integer and integral-valued float powers use exponentiation by squaring. Non-integral powers use a positive-real fast path where valid and otherwise the principal branch

[ z^x = \exp!\left(x\operatorname{Log}(z)\right), ]

where Log uses ln(abs(z)) and atan2(imaginary, real). Thus, for example, (-4) ** 0.5 has a positive imaginary component on the principal branch. 0 ** 0 is one, zero to a positive exponent is zero, and zero to a negative exponent raises a runtime domain error.