Simplification Rules

Addition

Example	Result
2 + x	x + 2
2 + (x + 2)	(x + 2) + 2
x + ax	ax + x
x + 0	x
0 + x	x
x + y	result of sum
x + x	2x
-y + x	x - y
x + (-y)	x - y
const + (const + x)	x + sum
const + (x + const)	x + sum
(const + x) + const	x + sum
(x + const) + const	x + sum
const + (const - x)	sum - x
const + (x - const)	diff + x
(const - x) + const	sum - x
(x - const) + const	diff + x
x + xb	(b + 1) * x
x + bx	(b + 1) * x
ax + x	(a + 1) * x
xa + x	(a + 1) * x
ax + bx	(a + b) * x
ax + xb	(a + b) * x
xa + bx	(a + b) * x
xa + xb	(a + b) * x

Note: italic result is precalculated.

Subtraction

Example	Result
0 - x	-x
x - 0	x
const - const	diff
x - x	0
x - (-y)	x + y
(const + x) - const	x + diff
(x + const) - const	x + diff
const - (const + x)	diff - x
const - (x + const)	diff - x
(const - x) - const	diff - x
(x - const) - const	x - sum
const - (const - x)	diff - x
const - (x - const)	sum - x
x - xb	(1 - b) * x
x - bx	(1 - b) * x
ax - x	(a - 1) * x
xa - x	(a - 1) * x
ax - bx	(a - b) * x
ax - xb	(a - b) * x
xa - bx	(a - b) * x
xa - xb	(a - b) * x

Note: italic result is precalculated.

Multiplication

Example	Result
xa	ax
x * ax	ax * x
2 * (2 * x)	(2 * x) * 2
0 * x	0
x * 0	0
1 * x	x
x * 1	x
-1 * x	-x
x * -1	-x
const * const	result of mul
x * -y	-(x * y)
x * x	2x
const * (const * x)	product * x
const * (x * const)	product * x
(const * x) * const	product * x
(x * const) * const	product * x
const * (const / x)	product / x
const * (x / const)	factor * x
(const / x) * const	product / x
(x / const) * const	factor * x
x * xb	b * x ^ 2
x * bx	b * x ^ 2
ax * x	a * x ^ 2
xa * x	a * x ^ 2
ax + bx	(a * b) * x ^ 2
ax + xb	(a * b) * x ^ 2
xa + bx	(a * b) * x ^ 2
xa + xb	(a * b) * x ^ 2
x * (1 / x)	1
(2 * x) * (1 / x)	2
(x * 2) * (1 / x)	2

Note: italic result is precalculated.

Division

Example	Result
0 / 0	NaN
0 / x	0
x / 0	DivideByZeroException
x / 1	x
const / const	result of div
x / x	1
(const * x) / const	x / fraction
(x * const) / const	x / fraction
const / (const * x)	fraction / x
const / (x * const)	fraction / x
(const / x) / const	fraction / x
(x / const) / const	x / mul
const / (const / x)	mul / x
const / (x / const)	mul / x

Note: italic result is precalculated.

Power

Example	Result
x^0	1
0^x	0
x^1	x
x ^ log(x, y)	y
e ^ ln(y)	y
10 ^ lg(y)	y
2 ^ lb(y)	y

Root

Example	Result
root(x, 1)	x

Exp

Example	Result
exp(ln(x))	x

Logarithm Lb, Lg, Ln, Log

Example	Result
lb(2)	1
lg(10)	1
ln(e)	1
log(x, x)	1

Unary Minus

Example	Result
-(-x)	x
-(number)	-number

Trigonometric

Example	Result
arccos(cos(x))	x
arccot(cot(x))	x
arccsc(csc(x))	x
arcsec(sec(x))	x
arcsin(sin(x))	x
arctan(tan(x))	x
cos(arccos(x))	x
cos(0)	1
cot(arccot(x))	x
cot(0)	+∞
csc(arccsc(x))	x
csc(0)	+∞
sec(arcsec(x))	x
sec(0)	1
sin(arcsin(x))	x
sin(0)	0
tan(arctan(x))	x
tan(0)	0

Hyperbolic

Example	Result
arcosh(cosh(x))	x
arcoth(coth(x))	x
arcsch(csch(x))	x
arsech(sech(x))	x
arsinh(sinh(x))	x
artanh(tanh(x))	x
cosh(accosh(x))	x
coth(accoth(x))	x
csch(accsch(x))	x
sech(acsech(x))	x
sinh(acsinh(x))	x
tanh(actanh(x))	x

ToDegree

Example	Result
todegree(number)	x degree
todegree(x degree)	x degree
todegree(angle number)	x degree

ToRadian

Example	Result
toradian(number)	x radian
toradian(x radian)	x radian
toradian(angle number)	x radian

ToGradian

Example	Result
togradian(number)	x gradian
togradian(x gradian)	x gradian
togradian(angle number)	x gradian

ToNumber

Example	Result
tonumber(angle number)	x

Abs

Example	Result
abs(-x)	x
abs(abs(x))	abs(x)