The Math
module contains module functions for basic trigonometric and transcendental functions. See class Float
for a list of constants that define Ruby's floating point accuracy.
Domains and codomains are given only for real (not complex) numbers.
Computes the arc cosine of x
. Returns 0..PI.
Domain: [-1, 1]
Codomain: [0, PI]
Math.acos(0) == Math::PI/2 #=> true
static VALUE math_acos(VALUE unused_obj, VALUE x) { double d; d = Get_Double(x); /* check for domain error */ if (d < -1.0 || 1.0 < d) domain_error("acos"); return DBL2NUM(acos(d)); }
Computes the inverse hyperbolic cosine of x
.
Domain: [1, INFINITY)
Codomain: [0, INFINITY)
Math.acosh(1) #=> 0.0
static VALUE math_acosh(VALUE unused_obj, VALUE x) { double d; d = Get_Double(x); /* check for domain error */ if (d < 1.0) domain_error("acosh"); return DBL2NUM(acosh(d)); }
Computes the arc sine of x
. Returns -PI/2..PI/2.
Domain: [-1, -1]
Codomain: [-PI/2, PI/2]
Math.asin(1) == Math::PI/2 #=> true
static VALUE math_asin(VALUE unused_obj, VALUE x) { double d; d = Get_Double(x); /* check for domain error */ if (d < -1.0 || 1.0 < d) domain_error("asin"); return DBL2NUM(asin(d)); }
Computes the inverse hyperbolic sine of x
.
Domain: (-INFINITY, INFINITY)
Codomain: (-INFINITY, INFINITY)
Math.asinh(1) #=> 0.881373587019543
static VALUE math_asinh(VALUE unused_obj, VALUE x) { return DBL2NUM(asinh(Get_Double(x))); }
Computes the arc tangent of x
. Returns -PI/2..PI/2.
Domain: (-INFINITY, INFINITY)
Codomain: (-PI/2, PI/2)
Math.atan(0) #=> 0.0
static VALUE math_atan(VALUE unused_obj, VALUE x) { return DBL2NUM(atan(Get_Double(x))); }
Computes the arc tangent given y
and x
. Returns a Float
in the range -PI..PI. Return value is a angle in radians between the positive x-axis of cartesian plane and the point given by the coordinates (x
, y
) on it.
Domain: (-INFINITY, INFINITY)
Codomain: [-PI, PI]
Math.atan2(-0.0, -1.0) #=> -3.141592653589793 Math.atan2(-1.0, -1.0) #=> -2.356194490192345 Math.atan2(-1.0, 0.0) #=> -1.5707963267948966 Math.atan2(-1.0, 1.0) #=> -0.7853981633974483 Math.atan2(-0.0, 1.0) #=> -0.0 Math.atan2(0.0, 1.0) #=> 0.0 Math.atan2(1.0, 1.0) #=> 0.7853981633974483 Math.atan2(1.0, 0.0) #=> 1.5707963267948966 Math.atan2(1.0, -1.0) #=> 2.356194490192345 Math.atan2(0.0, -1.0) #=> 3.141592653589793 Math.atan2(INFINITY, INFINITY) #=> 0.7853981633974483 Math.atan2(INFINITY, -INFINITY) #=> 2.356194490192345 Math.atan2(-INFINITY, INFINITY) #=> -0.7853981633974483 Math.atan2(-INFINITY, -INFINITY) #=> -2.356194490192345
static VALUE math_atan2(VALUE unused_obj, VALUE y, VALUE x) { double dx, dy; dx = Get_Double(x); dy = Get_Double(y); if (dx == 0.0 && dy == 0.0) { if (!signbit(dx)) return DBL2NUM(dy); if (!signbit(dy)) return DBL2NUM(M_PI); return DBL2NUM(-M_PI); } #ifndef ATAN2_INF_C99 if (isinf(dx) && isinf(dy)) { /* optimization for FLONUM */ if (dx < 0.0) { const double dz = (3.0 * M_PI / 4.0); return (dy < 0.0) ? DBL2NUM(-dz) : DBL2NUM(dz); } else { const double dz = (M_PI / 4.0); return (dy < 0.0) ? DBL2NUM(-dz) : DBL2NUM(dz); } } #endif return DBL2NUM(atan2(dy, dx)); }
Computes the inverse hyperbolic tangent of x
.
Domain: (-1, 1)
Codomain: (-INFINITY, INFINITY)
Math.atanh(1) #=> Infinity
static VALUE math_atanh(VALUE unused_obj, VALUE x) { double d; d = Get_Double(x); /* check for domain error */ if (d < -1.0 || +1.0 < d) domain_error("atanh"); /* check for pole error */ if (d == -1.0) return DBL2NUM(-INFINITY); if (d == +1.0) return DBL2NUM(+INFINITY); return DBL2NUM(atanh(d)); }
Returns the cube root of x
.
Domain: (-INFINITY, INFINITY)
Codomain: (-INFINITY, INFINITY)
-9.upto(9) {|x| p [x, Math.cbrt(x), Math.cbrt(x)**3] } #=> [-9, -2.0800838230519, -9.0] # [-8, -2.0, -8.0] # [-7, -1.91293118277239, -7.0] # [-6, -1.81712059283214, -6.0] # [-5, -1.7099759466767, -5.0] # [-4, -1.5874010519682, -4.0] # [-3, -1.44224957030741, -3.0] # [-2, -1.25992104989487, -2.0] # [-1, -1.0, -1.0] # [0, 0.0, 0.0] # [1, 1.0, 1.0] # [2, 1.25992104989487, 2.0] # [3, 1.44224957030741, 3.0] # [4, 1.5874010519682, 4.0] # [5, 1.7099759466767, 5.0] # [6, 1.81712059283214, 6.0] # [7, 1.91293118277239, 7.0] # [8, 2.0, 8.0] # [9, 2.0800838230519, 9.0]
static VALUE math_cbrt(VALUE unused_obj, VALUE x) { return DBL2NUM(cbrt(Get_Double(x))); }
Computes the cosine of x
(expressed in radians). Returns a Float
in the range -1.0..1.0.
Domain: (-INFINITY, INFINITY)
Codomain: [-1, 1]
Math.cos(Math::PI) #=> -1.0
static VALUE math_cos(VALUE unused_obj, VALUE x) { return DBL2NUM(cos(Get_Double(x))); }
Computes the hyperbolic cosine of x
(expressed in radians).
Domain: (-INFINITY, INFINITY)
Codomain: [1, INFINITY)
Math.cosh(0) #=> 1.0
static VALUE math_cosh(VALUE unused_obj, VALUE x) { return DBL2NUM(cosh(Get_Double(x))); }
Calculates the error function of x
.
Domain: (-INFINITY, INFINITY)
Codomain: (-1, 1)
Math.erf(0) #=> 0.0
static VALUE math_erf(VALUE unused_obj, VALUE x) { return DBL2NUM(erf(Get_Double(x))); }
Calculates the complementary error function of x.
Domain: (-INFINITY, INFINITY)
Codomain: (0, 2)
Math.erfc(0) #=> 1.0
static VALUE math_erfc(VALUE unused_obj, VALUE x) { return DBL2NUM(erfc(Get_Double(x))); }
Returns e**x.
Domain: (-INFINITY, INFINITY)
Codomain: (0, INFINITY)
Math.exp(0) #=> 1.0 Math.exp(1) #=> 2.718281828459045 Math.exp(1.5) #=> 4.4816890703380645
static VALUE math_exp(VALUE unused_obj, VALUE x) { return DBL2NUM(exp(Get_Double(x))); }
Returns a two-element array containing the normalized fraction (a Float
) and exponent (an Integer
) of x
.
fraction, exponent = Math.frexp(1234) #=> [0.6025390625, 11] fraction * 2**exponent #=> 1234.0
static VALUE math_frexp(VALUE unused_obj, VALUE x) { double d; int exp; d = frexp(Get_Double(x), &exp); return rb_assoc_new(DBL2NUM(d), INT2NUM(exp)); }
Calculates the gamma function of x.
Note that gamma(n) is same as fact(n-1) for integer n > 0. However gamma(n) returns float and can be an approximation.
def fact(n) (1..n).inject(1) {|r,i| r*i } end 1.upto(26) {|i| p [i, Math.gamma(i), fact(i-1)] } #=> [1, 1.0, 1] # [2, 1.0, 1] # [3, 2.0, 2] # [4, 6.0, 6] # [5, 24.0, 24] # [6, 120.0, 120] # [7, 720.0, 720] # [8, 5040.0, 5040] # [9, 40320.0, 40320] # [10, 362880.0, 362880] # [11, 3628800.0, 3628800] # [12, 39916800.0, 39916800] # [13, 479001600.0, 479001600] # [14, 6227020800.0, 6227020800] # [15, 87178291200.0, 87178291200] # [16, 1307674368000.0, 1307674368000] # [17, 20922789888000.0, 20922789888000] # [18, 355687428096000.0, 355687428096000] # [19, 6.402373705728e+15, 6402373705728000] # [20, 1.21645100408832e+17, 121645100408832000] # [21, 2.43290200817664e+18, 2432902008176640000] # [22, 5.109094217170944e+19, 51090942171709440000] # [23, 1.1240007277776077e+21, 1124000727777607680000] # [24, 2.5852016738885062e+22, 25852016738884976640000] # [25, 6.204484017332391e+23, 620448401733239439360000] # [26, 1.5511210043330954e+25, 15511210043330985984000000]
static VALUE math_gamma(VALUE unused_obj, VALUE x) { static const double fact_table[] = { /* fact(0) */ 1.0, /* fact(1) */ 1.0, /* fact(2) */ 2.0, /* fact(3) */ 6.0, /* fact(4) */ 24.0, /* fact(5) */ 120.0, /* fact(6) */ 720.0, /* fact(7) */ 5040.0, /* fact(8) */ 40320.0, /* fact(9) */ 362880.0, /* fact(10) */ 3628800.0, /* fact(11) */ 39916800.0, /* fact(12) */ 479001600.0, /* fact(13) */ 6227020800.0, /* fact(14) */ 87178291200.0, /* fact(15) */ 1307674368000.0, /* fact(16) */ 20922789888000.0, /* fact(17) */ 355687428096000.0, /* fact(18) */ 6402373705728000.0, /* fact(19) */ 121645100408832000.0, /* fact(20) */ 2432902008176640000.0, /* fact(21) */ 51090942171709440000.0, /* fact(22) */ 1124000727777607680000.0, /* fact(23)=25852016738884976640000 needs 56bit mantissa which is * impossible to represent exactly in IEEE 754 double which have * 53bit mantissa. */ }; enum {NFACT_TABLE = numberof(fact_table)}; double d; d = Get_Double(x); /* check for domain error */ if (isinf(d)) { if (signbit(d)) domain_error("gamma"); return DBL2NUM(INFINITY); } if (d == 0.0) { return signbit(d) ? DBL2NUM(-INFINITY) : DBL2NUM(INFINITY); } if (d == floor(d)) { if (d < 0.0) domain_error("gamma"); if (1.0 <= d && d <= (double)NFACT_TABLE) { return DBL2NUM(fact_table[(int)d - 1]); } } return DBL2NUM(tgamma(d)); }
Returns sqrt(x**2 + y**2), the hypotenuse of a right-angled triangle with sides x
and y
.
Math.hypot(3, 4) #=> 5.0
static VALUE math_hypot(VALUE unused_obj, VALUE x, VALUE y) { return DBL2NUM(hypot(Get_Double(x), Get_Double(y))); }
Returns the value of fraction
*(2**exponent
).
fraction, exponent = Math.frexp(1234) Math.ldexp(fraction, exponent) #=> 1234.0
static VALUE math_ldexp(VALUE unused_obj, VALUE x, VALUE n) { return DBL2NUM(ldexp(Get_Double(x), NUM2INT(n))); }
Calculates the logarithmic gamma of x
and the sign of gamma of x
.
Math.lgamma(x)
is same as
[Math.log(Math.gamma(x).abs), Math.gamma(x) < 0 ? -1 : 1]
but avoid overflow by Math.gamma(x)
for large x.
Math.lgamma(0) #=> [Infinity, 1]
static VALUE math_lgamma(VALUE unused_obj, VALUE x) { double d; int sign=1; VALUE v; d = Get_Double(x); /* check for domain error */ if (isinf(d)) { if (signbit(d)) domain_error("lgamma"); return rb_assoc_new(DBL2NUM(INFINITY), INT2FIX(1)); } if (d == 0.0) { VALUE vsign = signbit(d) ? INT2FIX(-1) : INT2FIX(+1); return rb_assoc_new(DBL2NUM(INFINITY), vsign); } v = DBL2NUM(lgamma_r(d, &sign)); return rb_assoc_new(v, INT2FIX(sign)); }
Returns the logarithm of x
. If additional second argument is given, it will be the base of logarithm. Otherwise it is e
(for the natural logarithm).
Domain: (0, INFINITY)
Codomain: (-INFINITY, INFINITY)
Math.log(0) #=> -Infinity Math.log(1) #=> 0.0 Math.log(Math::E) #=> 1.0 Math.log(Math::E**3) #=> 3.0 Math.log(12, 3) #=> 2.2618595071429146
static VALUE math_log(int argc, const VALUE *argv, VALUE unused_obj) { VALUE x, base; double d; rb_scan_args(argc, argv, "11", &x, &base); d = math_log1(x); if (argc == 2) { d /= math_log1(base); } return DBL2NUM(d); }
Returns the base 10 logarithm of x
.
Domain: (0, INFINITY)
Codomain: (-INFINITY, INFINITY)
Math.log10(1) #=> 0.0 Math.log10(10) #=> 1.0 Math.log10(10**100) #=> 100.0
static VALUE math_log10(VALUE unused_obj, VALUE x) { size_t numbits; double d = get_double_rshift(x, &numbits); /* check for domain error */ if (d < 0.0) domain_error("log10"); /* check for pole error */ if (d == 0.0) return DBL2NUM(-INFINITY); return DBL2NUM(log10(d) + numbits * log10(2)); /* log10(d * 2 ** numbits) */ }
Returns the base 2 logarithm of x
.
Domain: (0, INFINITY)
Codomain: (-INFINITY, INFINITY)
Math.log2(1) #=> 0.0 Math.log2(2) #=> 1.0 Math.log2(32768) #=> 15.0 Math.log2(65536) #=> 16.0
static VALUE math_log2(VALUE unused_obj, VALUE x) { size_t numbits; double d = get_double_rshift(x, &numbits); /* check for domain error */ if (d < 0.0) domain_error("log2"); /* check for pole error */ if (d == 0.0) return DBL2NUM(-INFINITY); return DBL2NUM(log2(d) + numbits); /* log2(d * 2 ** numbits) */ }
Computes the sine of x
(expressed in radians). Returns a Float
in the range -1.0..1.0.
Domain: (-INFINITY, INFINITY)
Codomain: [-1, 1]
Math.sin(Math::PI/2) #=> 1.0
static VALUE math_sin(VALUE unused_obj, VALUE x) { return DBL2NUM(sin(Get_Double(x))); }
Computes the hyperbolic sine of x
(expressed in radians).
Domain: (-INFINITY, INFINITY)
Codomain: (-INFINITY, INFINITY)
Math.sinh(0) #=> 0.0
static VALUE math_sinh(VALUE unused_obj, VALUE x) { return DBL2NUM(sinh(Get_Double(x))); }
Returns the non-negative square root of x
.
Domain: [0, INFINITY)
Codomain:[0, INFINITY)
0.upto(10) {|x| p [x, Math.sqrt(x), Math.sqrt(x)**2] } #=> [0, 0.0, 0.0] # [1, 1.0, 1.0] # [2, 1.4142135623731, 2.0] # [3, 1.73205080756888, 3.0] # [4, 2.0, 4.0] # [5, 2.23606797749979, 5.0] # [6, 2.44948974278318, 6.0] # [7, 2.64575131106459, 7.0] # [8, 2.82842712474619, 8.0] # [9, 3.0, 9.0] # [10, 3.16227766016838, 10.0]
Note that the limited precision of floating point arithmetic might lead to surprising results:
Math.sqrt(10**46).to_i #=> 99999999999999991611392 (!)
See also BigDecimal#sqrt and Integer.sqrt
.
static VALUE math_sqrt(VALUE unused_obj, VALUE x) { return rb_math_sqrt(x); }