// These transformations try to solve simple equations and teach Frink to // solve basic algebraic equations for the specified variable. // // For example, enter: // solve[3(x+y) === 10, x] // // (Right now this requires the triple-equals sign because Frink currently // requires that the left-hand-side of an assignment operator // ( = ) can actually be meaningfully assigned to, which may be // a constraint that needs to get loosened for temporary values. // This creates a named list of transformations called "solving" that we // can apply by name later. transformations solving { // Change sqrt[x] into a power. sqrt[_x] <-> _x^(1/2) // Move the variable we're solving for to the left side of the equation // if it's only on the right side of the equation. solve[_left === _right, _x] :: (freeOf[_left, _x] AND expressionContains[_right, _x]) <-> solve[_right === _left, _x] // Bailout condition solve[_x === _z, _x] :: freeOf[_z, _x] <-> _x === _z // Quadratic equations solve[(_a:1) _x^2 + (_b:1) _x === _c, _x] :: freeOf[_c, _x] <-> [ solve[_x === (-_b + (_b^2 - 4 _a (-_c))^(1/2)) / (2 _a), _x], solve[_x === (-_b - (_b^2 - 4 _a (-_c))^(1/2)) / (2 _a), _x] ] // Quadratic equation kludge. // Distributing something like (2x + 3p)(3x + 1) gives a result like // (6 x^2 + 9 p x + 2 x + 3p) which is a quadratic equation but has two // terms of x. This handles that common case in an inelegant way. // TODO: Find a better solution for this and remove it. solve[(_a:1) _x^2 + (_b1:1) _x + _b2 _x === _c, _x] :: freeOf[_c, _x] <-> [ solve[_x === (-(_b1 + _b2) + ((_b1 + _b2)^2 - 4 _a (-_c))^(1/2)) / (2 _a), _x],solve[_x === (-(_b1 + _b2) - ((_b1 + _b2)^2 - 4 _a (-_c))^(1/2)) / (2 _a), _x] ] // If both sides have an x, divide out x terms from right. // solve[_a === (_c:1) _x^(_b:1), _x] :: expressionContains[_a, _x] <-> solve[_a / _x^_b === _c, _x] // First move additive terms // Move all x-containing terms to left solve[_a === _c, _x] :: (expressionContains[_a, _x] && expressionContains[_c, _x]) <-> solve[_a - _c === 0, _x] // Move all non-x-containing terms to right. solve[_a + _b === _c, _x] :: (freeOf[_a,_x] AND expressionContains[_b, _x]) <-> solve[_b === _c - _a, _x] // Then move multiplicative terms. solve[_a * _b === _c, _x] :: (freeOf[_a,_x] AND expressionContains[_b, _x]) <-> solve[_b === _c / _a, _x] // Flip inverse exponents. solve[_a^_k is isNegative === _b, _c] :: expressionContains[_a, _c] <-> solve[_a^-_k === _b^-1, _c] // Solve for two terms containing c solve[_a _c + (_b:1) _c === _d, _c] :: freeOf[_a, _c] && freeOf[_b, _c] && freeOf[_d,_c] <-> solve[_c === _d / (_a + _b), _c] // Solve for negative and positive exponents on same side. solve[(_c1:1) _a^_k is isNegative + (_c2:1) _a^(_j:1) === _b, _c] :: expressionContains[_a, _c] && freeOf[_c1, _c] && freeOf[_c2, _c] <-> solve[_c1 + _c2 _a^(_j-_k) === _b _a^-_k, _c] // Very general negative and positive exponents on same side. solve[(_a:1) _x^_b is isNegative + _c === _d, _x] :: expressionContains[_c, _x] <-> solve[_a + _c _x^-_b === _d _x^-_b, _x] // Help the solver to factor an expression. solve[(_a:1) _x^_b + _c _x^_b === _d, _x] <-> solve[_x^_b === _d / (_a + _c), _x] // x in numerator and denominator (denominator has additive term.) solve[(_b:1) _x / ((_a:1) _x + _y) === _z, _x] :: freeOf[_b, _x] && freeOf[_a, _x] && freeOf[_y, _x] <-> solve[_x === _z (_a _x + _y) / _b, _x] // x in denominator of complicated fraction and outside fraction. solve[(_a:1) _x + (_b:1) / _d === _e, _x] :: expressionContains[_d, _x] <-> solve[_a _x _d + _b === _e _d, _x] // Solve for squared terms. // Results are a list of two different solutions. solve[_a^_k is isPositive === _b, _c] :: expressionContains[_a, _c] AND (_k mod 2 == 0) <-> [ solve[_a^(_k/2) === _b^(1/2),_c] , solve[_a^(_k/2) === -_b^(1/2), _c ] ] // a x + b (d+ (c x)^(1/2)) === z // solve[(_a:1) _x + (_b:1) ((_d:0) + ((_c:1) _x)^(1/2)) === _z, _x] :: freeOf[_a, _x] and freeOf[_b,_x] and freeOf[_c, _x] and freeOf[_d, _x] and freeOf[_z, _x] <-> [ solve[sqrt[-4 _a _b^3 _c _d + 4 _a _b^2 _c _z + _b^4 _c^2] - 2 _a _b _d + 2 _a _z + _b^2 _c, _x], solve[-sqrt[-4 _a _b^3 _c _d + 4 _a _b^2 _c _z + _b^4 _c^2] - 2 _a _b _d + 2 _a _z + _b^2 _c, _x] ] // Force grouping of terms together. Note that this is the inverse of the // distributive transform below and could cause loops. solve[(_a:1) _x + (_b:1) _x + (_c:0), _x] <-> solve[_x (_a + _b) + _c, _x] // a x^(1/2) + b x== d // solve[((_a:1) _x^(1/2)) + (_b:1) _x === _d, _x] :: freeOf[_a, _x] and freeOf[_b,_x] and freeOf[_d, _x] <-> [ solve[_x === (_a sqrt[_a^2 + 4 _b _d] + _a^2 + 2 _b _d)/(2 _b^2), _x], solve[_x === (-_a sqrt[_a^2 + 4 _b _d] + _a^2 + 2 _b _d)/(2 _b^2), _x]] // General solving help when a term includes x + x^(1/2) terms. // Shift the x^(1/2) term to one side of the equals sign and square both // sides. This generally lets it be solved by the quadratic equation. solve[_a + (_d:1) _b^(1/2) === _c, _x] :: expressionContains[_a, _x] and expressionContains[_b, _x] and freeOf[_c, _x] and freeOf[_d, _x] <-> solve[(_a - _c)^2 === _d^2 _b, _x] // General solving help when a term includes x * x^(1/2) terms. // Shift the x^(1/2) term to one side of the equals sign and square both // sides. solve[_a _b^(1/2) + (_d:0) === _c, _x] :: expressionContains[_a, _x] and expressionContains[_b, _x] and freeOf[_c, _x] <-> solve[_b _a^2 === (_c-_d)^2, _x] // General solving help when a term includes x * x^(-1/2) terms. // Shift the x^(-1/2) term to one side of the equals sign and square both // sides. solve[_a _b^(-1/2) + (_d:0) === _c, _x] :: expressionContains[_a, _x] and expressionContains[_b, _x] and freeOf[_c, _x] <-> solve[_b === _a^2 / (_c-_d)^2, _x] // General solving help when a term contains a x + b (c + d x)^2 terms. // Expand out (c + d x) which usually allows the equation to be solved // with the quadratic equation. // TODO: Generalize this for cubed terms, etc solve[(_a:1) + (_b:1) (_c + (_d:1) _x)^2 === _f, _x] :: expressionContains[_a, _x] and freeOf[_c, _x] and freeOf[_f, _x] <-> solve[_a + _b (_c^2 + 2 (_c _d _x) + _d^2 _x^2) === _f, _x] // Solving help with two fractions with denominators containing x. // THIS CREATES AN INFINITE LOOP SOMEWHERE // solve[(_a:1) / _b + (_c:1) / _d === _f, _x] :: expressionContains[_b, _x] and expressionContains[_d, _x] and freeOf[_a, _x] and freeOf[_c, _x] <-> solve[(_a _d + _b _c) / (_b _d) === _f, _x] // a x (b + c x^2)^(-1/2) == d // solve[(_a:1) _x (_b + (_c:1) _x^2)^(-1/2) === _d, _x] <-> [solve[_x === i _b^(1/2) _d / (_c _d^2 - _a^2)^(1/2), _x], solve[_x === -i _b^(1/2) _d / (_c _d^2 - _a^2)^(1/2), _x]] // _a + (_b + _x)^2 == d with a containing x. Multiply out the parens. solve[_a + (_b + (_c:1) _x)^2 === _d, _x] :: expressionContains[_a, _x] <-> solve[_a + _b^2 + 2 _b _c _x + _c^2 _x^2 === _d, _x] // Factor out 3 terms. TODO: Generalize this! solve[(_a:1) _x + _b _x + _c _x === _d, _x] <-> solve[(_a + _b + _c) _x === _d, _x] // Solve for powers of 3, 6, 9, etc. solve[_a^_k is isPositive === _b, _c] :: expressionContains[_a, _c] AND (_k mod 3 == 0) <-> [ solve[_a^(_k/3) === _b^(1/3),_c] , solve[_a^(_k/3) === -((-1)^(1./3)) _b^(1/3), _c ], solve[_a^(_k/3) === ((-1)^(2./3)) _b^(1/3), _c ] ] // Solve for rational exponents solve[_a^_k is isRational === _b, _c] :: expressionContains[_a, _c] <-> solve[_a === _b^(1/_k),_c] // Gah! Cubic equations! // See http://en.wikipedia.org/wiki/Cubic_function#General_formula_of_roots // // See https://brilliant.org/wiki/cubic-discriminant/ // The discriminant of a x^3 + b x^2 + c x == d is // delta = b^2 c^2 - 4 a c^3 + 4b^3 d - 27 a^2 d^2 - 18 a b c d // (note that this is different than the usual discriminant equation // because Frink's solvers will put d on the right-hand side of the // equation so d is replaced with -d) // // If delta > 0 then the equation has three distinct real roots. // if delta == 0 then the equation has a repeated root and all its roots are real. // if delta < 0 then the equation has one real root and two non-real complex conjugate roots. (which is currently what is solved below.) // // TODO: Find a way to store repeated temporary parts of results into variables. // Solving when a, b, and c are all defined. solve[(_a:1) _x^3 + (_b:1) _x^2 + (_c:1) _x === _d, _x] :: freeOf[_d, _x] <-> [solve[_x === -_b/(3 _a) - (2^(1/3)*(-_b^2 + 3 _a _c))/ (3 _a (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d + sqrt[-4 (_b^2 - 3 _a _c)^3 + (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d)^2])^(1/3)) + (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d + sqrt[-4 (_b^2 - 3 _a _c)^3 + (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d)^2])^(1/3)/(3*2^(1/3)*_a), _x], solve[_x === -_b/(3 _a) + ((1 + i sqrt[3])*(-_b^2 + 3 _a _c))/ (3*2^(2/3) _a (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d + sqrt[-4 (_b^2 - 3 _a _c)^3 + (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d)^2])^(1/3)) - ((1 - i sqrt[3])*(-2 _b^3 + 9 _a _b _c + 27 _a^2 _d + sqrt[-4 (_b^2 - 3 _a _c)^3 + (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d)^2])^(1/3)) / (6*2^(1/3)*_a), _x], solve[_x === -_b/(3 _a) + ((1 - i sqrt[3])*(-_b^2 + 3 _a _c))/ (3*2^(2/3)*_a*(-2 _b^3 + 9 _a _b _c + 27 _a^2 _d + sqrt[-4 (_b^2 - 3 _a _c)^3 + (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d)^2])^(1/3)) - ((1 + i sqrt[3])*(-2 _b^3 + 9 _a _b _c + 27 _a^2 _d + sqrt[-4 (_b^2 - 3 _a _c)^3 + (-2 _b^3 + 9 _a _b _c + 27 _a^2 _d)^2])^(1/3)) / (6*2^(1/3)*_a), _x]] // Cubic equation solver when b = 0 (coefficient of x^2) solve[(_a:1) _x^3 + (_c:1) _x === _d, _x] :: freeOf[_d, _x] <-> [solve[_x === 0 - (2^(1/3)*(3 _a _c))/ (3 _a (27 _a^2 _d + sqrt[-4 (-3 _a _c)^3 + (27 _a^2 _d)^2])^(1/3)) + (27 _a^2 _d + sqrt[-4 (-3 _a _c)^3 + (27 _a^2 _d)^2])^(1/3)/(3*2^(1/3)*_a), _x], solve[_x === ((1 + i sqrt[3])*(3 _a _c))/ (3*2^(2/3) _a (27 _a^2 _d + sqrt[-4 (-3 _a _c)^3 + (27 _a^2 _d)^2])^(1/3)) - ((1 - i sqrt[3])*(27 _a^2 _d + sqrt[-4 (-3 _a _c)^3 + (27 _a^2 _d)^2])^(1/3)) / (6*2^(1/3)*_a), _x], solve[_x === ((1 - i sqrt[3])*(3 _a _c))/ (3*2^(2/3)*_a*(27 _a^2 _d + sqrt[-4 (3 _a _c)^3 + (27 _a^2 _d)^2])^(1/3)) - ((1 + i sqrt[3])*(27 _a^2 _d + sqrt[-4 (-3 _a _c)^3 + (27 _a^2 _d)^2])^(1/3)) / (6*2^(1/3)*_a), _x]] // Cubic equation solver when c=0 (coefficient of x) solve[(_a:1) _x^3 + (_b:1) _x^2 === _d, _x] :: freeOf[_d, _x] <-> [solve[_x === -_b/(3 _a) - (2^(1/3)*(-_b^2))/ (3 _a (-2 _b^3 + 27 _a^2 _d + sqrt[-4 (_b^2)^3 + (-2 _b^3 + 27 _a^2 _d)^2])^(1/3)) + (-2 _b^3 + 27 _a^2 _d + sqrt[-4 (_b^2)^3 + (-2 _b^3 + 27 _a^2 _d)^2])^(1/3)/(3*2^(1/3)*_a), _x], solve[_x === -_b/(3 _a) + ((1 + i sqrt[3])*(-_b^2))/ (3*2^(2/3) _a (-2 _b^3 + 27 _a^2 _d + sqrt[-4 (_b^2)^3 + (-2 _b^3 + 27 _a^2 _d)^2])^(1/3)) - ((1 - i sqrt[3])*(-2 _b^3 + 27 _a^2 _d + sqrt[-4 (_b^2)^3 + (-2 _b^3 + 27 _a^2 _d)^2])^(1/3)) / (6*2^(1/3)*_a), _x], solve[_x === -_b/(3 _a) + ((1 - i sqrt[3])*(-_b^2))/ (3*2^(2/3)*_a*(-2 _b^3 + 27 _a^2 _d + sqrt[-4 (_b^2)^3 + (-2 _b^3 + 27 _a^2 _d)^2])^(1/3)) - ((1 + i sqrt[3])*(-2 _b^3 + + 27 _a^2 _d + sqrt[-4 (_b^2)^3 + (-2 _b^3 + 27 _a^2 _d)^2])^(1/3)) / (6*2^(1/3)*_a), _x]] // Cubic equation solver // See https://sciencing.com/solve-cubic-equations-8136094.html // x = (q + (q^2 + (r-p^2)^3)^(1/2))^(1/3) + (q-(q^2+(r-p^2)^3)^(1/2))^(1/3) + p // p = -b/(3a) // q = p^3 + (b c - 3 a d)/(6 a^2) // r = c / (3 a) // Replace floating-point approximation to zero with integer 0. 0. <-> 0 // Some simplifying rules that actually aren't appropriate if you're // tracking units. These are not really valid because 0 feet != 0 days // and 0 feet + 0 is not units-correct. // TODO: FIX THIS! We need a pattern that matches units with magnitude // of zero. 0 _x <-> 0 // 0 + _x <-> _x 1 _x <-> _x 1^_x <-> 1 ln[e] <-> 1 ln[1] <-> 0 log[10] <-> 1 log[1] <-> 0 log[_a, _b] <-> ln[_a] / ln[_b] // This transforms rules in terms of log base 10 into equations of // natural log. This makes a lot of different types of equations more // readily solved by already-existing rules: for example, LambertW rules, // (see powerTransformations.frink) // and eliminates the need to rewrite all of these rules in multiple forms. log[_a] <-> ln[_a] / ln[10] // Simplifying rule e^ln[_x] <-> _x // Simplifying rule e^(_a ln[_b]) <-> _b^_a // Simplifying rule ln[_a ^ _b] <-> _b ln[_a] // Simplifying rule log[_a ^ _b] <-> _b log[_a] // Exponentiate out parts. This is only valid if the exponent is an integer (_a _b)^_c :: isInteger[_c] <-> _a^_c _b^_c // Distribute (to often clarify and simplify) // (Note: this is often disadvantageous when using // interval arguments as intervals are subdistributive and the result // may be wider.) _a (_c + _d) <-> _a _c + _a _d // Combine coefficients of x; // This may cause loops with the above expression. // _a _x + _b _x + (_c:0) <-> (_a + _b) _x + _c // (_a + _b)^_k :: isInteger[_k] AND _k >= 2 <-> (_a^2 + 2 _a _b + _b^2)(_a + _b)^(_k-2) // Sinc function sinc[_x] <-> _x == 0 ? 1 : sin[_x]/_x sinc[0] <-> 1 // Simplify trinary conditionals _x == _x <-> true true ? _x : _y <-> _x false ? _x : _y <-> _y } "solvingTransformations.frink included ok!"