Math

Math 模块

当需要mathn时，Math模块更改如下：

标准数学模块行为：

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

在需要'mathn'之后，这个更改为：

require 'mathn'
Math.sqrt(4/9)      # => 2/3
Math.sqrt(4.0/9.0)  # => 0.666666666666667
Math.sqrt(- 4/9)    # => Complex(0, 2/3)

数学模块包含基本三角函数和超越函数的模块函数。请参阅Float类以获取定义Ruby浮点精度的常量列表。

域和codomains仅适用于真实（不复杂）的数字。

常量

数学常数E（e）定义为浮点数。

数学常量PI定义为浮点数。

公共类方法

acos(x) → Float Show source

计算x的反余弦。返回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)
}

acosh(x) → Float Show source

计算x的反双曲余弦。

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)
}

asin(x) → Float Show source

计算x的反正弦。返回-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)
}

asinh(x) → Float Show source

计算x的反双曲正弦。

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))
}

atan(x) → Float Show source

计算x的反正切。返回-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))
}

atan2(y, x) → Float Show source

计算给定y和x的反正切。返回范围-PI..PI中的Float。返回值是笛卡尔平面的正x轴与其坐标（x，y）给出的点之间的角度弧度。

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)
}

atanh(x) → Float Show source

计算x的反双曲正切。

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)
}

cbrt(x) → Float Show source

返回x的立方体根。

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))
}

cos(x) → Float Show source

计算x的余弦（以弧度表示）。返回-1.0..1.0范围内的Float值。

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))
}

cosh(x) → Float Show source

计算x的双曲余弦（以弧度表示）。

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))
}

erf(x) → Float Show source

计算x的误差函数。

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))
}

erfc(x) → Float Show source

计算x的互补误差函数。

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))
}

exp(x) → Float Show source

返回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))
}

frexp(x) → fraction, exponent()

返回包含归一化分数（Float）和指数（Integer）的两元素数组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)
}

gamma(x) → Float Show source

计算x的伽玛函数。

请注意，对于整数n> 0，gamma（n）与fact（n-1）相同。但是gamma（n）返回float并且可以是近似值。

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) && signbit(d)) domain_error("gamma"
    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)
}

hypot(x, y) → Float Show source

返回sqrt（x ** 2 + y ** 2），即边x和y的直角三角形的斜边。

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))
}

ldexp(fraction, exponent) → float Show source

返回fraction*（2 ** exponent）的值。

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))
}

lgamma(x) → float, -1 or 1()

计算x的对数伽马和x的伽马符号。

:: lgamma与以下形式相同：

[Math.log(Math.gamma(x).abs), Math.gamma(x) < 0 ? -1 : 1]

但这是为了避免大的x的 :: gamma溢出。

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)
    }
    v = DBL2NUM(lgamma_r(d, &sign)
    return rb_assoc_new(v, INT2FIX(sign)
}

log(x) → Float Show source

log(x, base) → Float

返回x的对数。如果再给出第二个参数，它将成为对数的基础。否则它是e（对于自然对数）。

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
}

log10(x) → Float Show source

返回x的基数10的对数。

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) */
}

log2(x) → Float Show source

返回x的基数2的对数。

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) */
}

rsqrt(a) Show source

计算非负数的平方根。此方法由Math.sqrt进行内部使用。

# File lib/mathn.rb, line 119
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

sin(x) → Float Show source

计算x的正弦值（以弧度表示）。返回-1.0..1.0范围内的Float。

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))
}

sinh(x) → Float Show source

计算x的双曲正弦（以弧度表示）。

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))
}

sqrt(a) Show source

计算a的平方根。如果可能的话，它使用Complex和Rational来避免舍入错误。

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 103
def sqrt(a)
  if a.kind_of?(Complex)
    sqrt!(a)
  elsif a.respond_to?(:nan?) and a.nan?
    a
  elsif a >= 0
    rsqrt(a)
  else
    Complex(0,rsqrt(-a))
  end
end

tan(x) → Float Show source

计算x的正切（以弧度表示）。

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.tan(0) #=> 0.0

static VALUE
math_tan(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(tan(Get_Double(x))
}

tanh(x) → Float Show source

计算x的双曲正切（以弧度表示）。

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

Math.tanh(0) #=> 0.0

static VALUE
math_tanh(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(tanh(Get_Double(x))
}

私有实例方法

rsqrt(a) Show source

计算非负数的平方根。此方法由Math.sqrt内部使用。

# File lib/mathn.rb, line 119
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

sqrt(a) Show source

计算a的平方根。如果可能的话，它使用Complex和Rational来避免舍入错误。

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 103
def sqrt(a)
  if a.kind_of?(Complex)
    sqrt!(a)
  elsif a.respond_to?(:nan?) and a.nan?
    a
  elsif a >= 0
    rsqrt(a)
  else
    Complex(0,rsqrt(-a))
  end
end