module Math
When mathn is required, the Math module changes as follows:
Standard Math module behaviour:
Math.sqrt(4/9) # => 0.0 Math.sqrt(4.0/9.0) # => 0.666666666666667 Math.sqrt(- 4/9) # => Errno::EDOM: Numerical argument out of domain - sqrt
After require 'mathn', this is changed to:
require 'mathn' Math.sqrt(4/9) # => 2/3 Math.sqrt(4.0/9.0) # => 0.666666666666667 Math.sqrt(- 4/9) # => Complex(0, 2/3)
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.
Constants
Public Class Methods
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 obj, VALUE x) { double d0, d; Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (d0 < -1.0 || 1.0 < d0) domain_error("acos"); d = acos(d0); return DBL2NUM(d); }
Computes the inverse hyperbolic cosine of x
.
Domain: [1, INFINITY)
Codomain: [0, INFINITY)
Math.acosh(1) #=> 0.0
static VALUE math_acosh(VALUE obj, VALUE x) { double d0, d; Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (d0 < 1.0) domain_error("acosh"); d = acosh(d0); return DBL2NUM(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 obj, VALUE x) { double d0, d; Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (d0 < -1.0 || 1.0 < d0) domain_error("asin"); d = asin(d0); return DBL2NUM(d); }
Computes the inverse hyperbolic sine of x
.
Domain: (-INFINITY, INFINITY)
Codomain: (-INFINITY, INFINITY)
Math.asinh(1) #=> 0.881373587019543
static VALUE math_asinh(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(asinh(RFLOAT_VALUE(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 obj, VALUE x) { Need_Float(x); return DBL2NUM(atan(RFLOAT_VALUE(x))); }
Computes the arc tangent given y
and x
. Returns a
Float in the range -PI..PI.
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 obj, VALUE y, VALUE x) { #ifndef M_PI # define M_PI 3.14159265358979323846 #endif double dx, dy; Need_Float2(y, x); dx = RFLOAT_VALUE(x); dy = RFLOAT_VALUE(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 obj, VALUE x) { double d0, d; Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (d0 < -1.0 || +1.0 < d0) domain_error("atanh"); /* check for pole error */ if (d0 == -1.0) return DBL2NUM(-INFINITY); if (d0 == +1.0) return DBL2NUM(+INFINITY); d = atanh(d0); return DBL2NUM(d); }
Returns the cube root of x
.
Domain: [0, INFINITY)
Codomain:[0, 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 obj, VALUE x) { Need_Float(x); return DBL2NUM(cbrt(RFLOAT_VALUE(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 obj, VALUE x) { Need_Float(x); return DBL2NUM(cos(RFLOAT_VALUE(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 obj, VALUE x) { Need_Float(x); return DBL2NUM(cosh(RFLOAT_VALUE(x))); }
Calculates the error function of x
.
Domain: (-INFINITY, INFINITY)
Codomain: (-1, 1)
Math.erf(0) #=> 0.0
static VALUE math_erf(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(erf(RFLOAT_VALUE(x))); }
Calculates the complementary error function of x.
Domain: (-INFINITY, INFINITY)
Codomain: (0, 2)
Math.erfc(0) #=> 1.0
static VALUE math_erfc(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(erfc(RFLOAT_VALUE(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 obj, VALUE x) { Need_Float(x); return DBL2NUM(exp(RFLOAT_VALUE(x))); }
Returns a two-element array containing the normalized fraction (a Float) and exponent (a Fixnum) of x
.
fraction, exponent = Math.frexp(1234) #=> [0.6025390625, 11] fraction * 2**exponent #=> 1234.0
static VALUE math_frexp(VALUE obj, VALUE x) { double d; int exp; Need_Float(x); d = frexp(RFLOAT_VALUE(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 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. */ }; double d0, d; double intpart, fracpart; Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (isinf(d0) && signbit(d0)) domain_error("gamma"); fracpart = modf(d0, &intpart); if (fracpart == 0.0) { if (intpart < 0) domain_error("gamma"); if (0 < intpart && intpart - 1 < (double)numberof(fact_table)) { return DBL2NUM(fact_table[(int)intpart - 1]); } } d = tgamma(d0); return DBL2NUM(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 obj, VALUE x, VALUE y) { Need_Float2(x, y); return DBL2NUM(hypot(RFLOAT_VALUE(x), RFLOAT_VALUE(y))); }
Returns the value of fraction
*(2**exponent
).
fraction, exponent = Math.frexp(1234) Math.ldexp(fraction, exponent) #=> 1234.0
static VALUE math_ldexp(VALUE obj, VALUE x, VALUE n) { Need_Float(x); return DBL2NUM(ldexp(RFLOAT_VALUE(x), NUM2INT(n))); }
Calculates the logarithmic gamma of x
and the sign of gamma of
x
.
::lgamma is same as
[Math.log(Math.gamma(x).abs), Math.gamma(x) < 0 ? -1 : 1]
but avoid overflow by ::gamma for large x.
Math.lgamma(0) #=> [Infinity, 1]
static VALUE math_lgamma(VALUE obj, VALUE x) { double d0, d; int sign=1; VALUE v; Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (isinf(d0)) { if (signbit(d0)) domain_error("lgamma"); return rb_assoc_new(DBL2NUM(INFINITY), INT2FIX(1)); } d = lgamma_r(d0, &sign); v = DBL2NUM(d); 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 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 obj, VALUE x) { double d0, d; size_t numbits; if (RB_BIGNUM_TYPE_P(x) && BIGNUM_POSITIVE_P(x) && DBL_MAX_EXP <= (numbits = rb_absint_numwords(x, 1, NULL))) { numbits -= DBL_MANT_DIG; x = rb_big_rshift(x, SIZET2NUM(numbits)); } else { numbits = 0; } Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (d0 < 0.0) domain_error("log10"); /* check for pole error */ if (d0 == 0.0) return DBL2NUM(-INFINITY); d = log10(d0); if (numbits) d += numbits * log10(2); /* log10(2**numbits) */ return DBL2NUM(d); }
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 obj, VALUE x) { double d0, d; size_t numbits; if (RB_BIGNUM_TYPE_P(x) && BIGNUM_POSITIVE_P(x) && DBL_MAX_EXP <= (numbits = rb_absint_numwords(x, 1, NULL))) { numbits -= DBL_MANT_DIG; x = rb_big_rshift(x, SIZET2NUM(numbits)); } else { numbits = 0; } Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (d0 < 0.0) domain_error("log2"); /* check for pole error */ if (d0 == 0.0) return DBL2NUM(-INFINITY); d = log2(d0); d += numbits; return DBL2NUM(d); }
Compute square root of a non negative number. This method is internally
used by Math.sqrt
.
# File lib/mathn.rb, line 141 def rsqrt(a) if a.kind_of?(Float) sqrt!(a) elsif a.kind_of?(Rational) rsqrt(a.numerator)/rsqrt(a.denominator) else src = a max = 2 ** 32 byte_a = [src & 0xffffffff] # ruby's bug while (src >= max) and (src >>= 32) byte_a.unshift src & 0xffffffff end answer = 0 main = 0 side = 0 for elm in byte_a main = (main << 32) + elm side <<= 16 if answer != 0 if main * 4 < side * side applo = main.div(side) else applo = ((sqrt!(side * side + 4 * main) - side)/2.0).to_i + 1 end else applo = sqrt!(main).to_i + 1 end while (x = (side + applo) * applo) > main applo -= 1 end main -= x answer = (answer << 16) + applo side += applo * 2 end if main == 0 answer else sqrt!(a) end end end
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 obj, VALUE x) { Need_Float(x); return DBL2NUM(sin(RFLOAT_VALUE(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 obj, VALUE x) { Need_Float(x); return DBL2NUM(sinh(RFLOAT_VALUE(x))); }
Computes the square root of a
. It makes use of Complex and Rational to
have no rounding errors if possible.
Math.sqrt(4/9) # => 2/3 Math.sqrt(- 4/9) # => Complex(0, 2/3) Math.sqrt(4.0/9.0) # => 0.666666666666667
# File lib/mathn.rb, line 118 def sqrt(a) if a.kind_of?(Complex) abs = sqrt(a.real*a.real + a.imag*a.imag) x = sqrt((a.real + abs)/Rational(2)) y = sqrt((-a.real + abs)/Rational(2)) if a.imag >= 0 Complex(x, y) else Complex(x, -y) end elsif a.respond_to?(:nan?) and a.nan? a elsif a >= 0 rsqrt(a) else Complex(0,rsqrt(-a)) end end
Computes the tangent of x
(expressed in radians).
Domain: (-INFINITY, INFINITY)
Codomain: (-INFINITY, INFINITY)
Math.tan(0) #=> 0.0
static VALUE math_tan(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(tan(RFLOAT_VALUE(x))); }
Computes the hyperbolic tangent of x
(expressed in radians).
Domain: (-INFINITY, INFINITY)
Codomain: (-1, 1)
Math.tanh(0) #=> 0.0
static VALUE math_tanh(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(tanh(RFLOAT_VALUE(x))); }
Private Instance Methods
Compute square root of a non negative number. This method is internally
used by Math.sqrt
.
# File lib/mathn.rb, line 141 def rsqrt(a) if a.kind_of?(Float) sqrt!(a) elsif a.kind_of?(Rational) rsqrt(a.numerator)/rsqrt(a.denominator) else src = a max = 2 ** 32 byte_a = [src & 0xffffffff] # ruby's bug while (src >= max) and (src >>= 32) byte_a.unshift src & 0xffffffff end answer = 0 main = 0 side = 0 for elm in byte_a main = (main << 32) + elm side <<= 16 if answer != 0 if main * 4 < side * side applo = main.div(side) else applo = ((sqrt!(side * side + 4 * main) - side)/2.0).to_i + 1 end else applo = sqrt!(main).to_i + 1 end while (x = (side + applo) * applo) > main applo -= 1 end main -= x answer = (answer << 16) + applo side += applo * 2 end if main == 0 answer else sqrt!(a) end end end
Computes the square root of a
. It makes use of Complex and Rational to
have no rounding errors if possible.
Math.sqrt(4/9) # => 2/3 Math.sqrt(- 4/9) # => Complex(0, 2/3) Math.sqrt(4.0/9.0) # => 0.666666666666667
# File lib/mathn.rb, line 118 def sqrt(a) if a.kind_of?(Complex) abs = sqrt(a.real*a.real + a.imag*a.imag) x = sqrt((a.real + abs)/Rational(2)) y = sqrt((-a.real + abs)/Rational(2)) if a.imag >= 0 Complex(x, y) else Complex(x, -y) end elsif a.respond_to?(:nan?) and a.nan? a elsif a >= 0 rsqrt(a) else Complex(0,rsqrt(-a)) end end