Module: Math

Defined in:
math.c

Defined Under Namespace

Classes: DomainError

Constant Summary collapse

PI =

Definition of the mathematical constant PI as a Float number.

DBL2NUM(M_PI)
E =

Definition of the mathematical constant E for Euler’s number (e) as a Float number.

DBL2NUM(exp(1.0))

Class Method Summary collapse

Instance Method Summary collapse

Class Method Details

.acos(x) ⇒ Float

Computes the arc cosine of x. Returns 0..PI.

Domain: [-1, 1]

Codomain: [0, PI]

Math.acos(0) == Math::PI/2  #=> true

Returns:



172
173
174
175
176
177
178
179
180
181
 
# File 'math.c', line 172

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

Computes the inverse hyperbolic cosine of x.

Domain: [1, INFINITY)

Codomain: [0, INFINITY)

Math.acosh(1) #=> 0.0

Returns:



331
332
333
334
335
336
337
338
339
340
 
# File 'math.c', line 331

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

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

Returns:



196
197
198
199
200
201
202
203
204
205
 
# File 'math.c', line 196

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

Computes the inverse hyperbolic sine of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.asinh(1) #=> 0.881373587019543

Returns:



356
357
358
359
360
 
# File 'math.c', line 356

static VALUE
math_asinh(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(asinh(Get_Double(x)));
}

.atan(x) ⇒ Float

Computes the arc tangent of x. Returns -PI/2..PI/2.

Domain: (-INFINITY, INFINITY)

Codomain: (-PI/2, PI/2)

Math.atan(0) #=> 0.0

Returns:



220
221
222
223
224
 
# File 'math.c', line 220

static VALUE
math_atan(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(atan(Get_Double(x)));
}

.atan2(y, x) ⇒ Float

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

Returns:



65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
 
# File 'math.c', line 65

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

Computes the inverse hyperbolic tangent of x.

Domain: (-1, 1)

Codomain: (-INFINITY, INFINITY)

Math.atanh(1) #=> Infinity

Returns:



376
377
378
379
380
381
382
383
384
385
386
387
388
 
# File 'math.c', line 376

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(-HUGE_VAL);
    if (d == +1.0) return DBL2NUM(+HUGE_VAL);
    return DBL2NUM(atanh(d));
}

.cbrt(x) ⇒ Float

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]

Returns:



691
692
693
694
695
696
697
698
699
700
701
702
 
# File 'math.c', line 691

static VALUE
math_cbrt(VALUE unused_obj, VALUE x)
{
    double f = Get_Double(x);
    double r = cbrt(f);
#if defined __GLIBC__
    if (isfinite(r)) {
	r = (2.0 * r + (f / r / r)) / 3.0;
    }
#endif
    return DBL2NUM(r);
}

.cos(x) ⇒ Float

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

Returns:



110
111
112
113
114
 
# File 'math.c', line 110

static VALUE
math_cos(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(cos(Get_Double(x)));
}

.cosh(x) ⇒ Float

Computes the hyperbolic cosine of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: [1, INFINITY)

Math.cosh(0) #=> 1.0

Returns:



248
249
250
251
252
 
# File 'math.c', line 248

static VALUE
math_cosh(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(cosh(Get_Double(x)));
}

.erf(x) ⇒ Float

Calculates the error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

  Math.erf(0) #=> 0.0

Returns:



771
772
773
774
775
 
# File 'math.c', line 771

static VALUE
math_erf(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(erf(Get_Double(x)));
}

.erfc(x) ⇒ Float

Calculates the complementary error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (0, 2)

  Math.erfc(0) #=> 1.0

Returns:



791
792
793
794
795
 
# File 'math.c', line 791

static VALUE
math_erfc(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(erfc(Get_Double(x)));
}

.exp(x) ⇒ Float

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

Returns:



406
407
408
409
410
 
# File 'math.c', line 406

static VALUE
math_exp(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(exp(Get_Double(x)));
}

.frexp(x) ⇒ Array

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

Returns:



715
716
717
718
719
720
721
722
723
 
# File 'math.c', line 715

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

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]

Returns:



837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
 
# File 'math.c', line 837

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(HUGE_VAL);
    }
    if (d == 0.0) {
	return signbit(d) ? DBL2NUM(-HUGE_VAL) : DBL2NUM(HUGE_VAL);
    }
    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

Returns sqrt(x**2 + y**2), the hypotenuse of a right-angled triangle with sides x and y.

Math.hypot(3, 4)   #=> 5.0

Returns:



751
752
753
754
755
 
# File 'math.c', line 751

static VALUE
math_hypot(VALUE unused_obj, VALUE x, VALUE y)
{
    return DBL2NUM(hypot(Get_Double(x), Get_Double(y)));
}

.ldexp(fraction, exponent) ⇒ Float

Returns the value of fraction*(2**exponent).

fraction, exponent = Math.frexp(1234)
Math.ldexp(fraction, exponent)   #=> 1234.0

Returns:



735
736
737
738
739
 
# File 'math.c', line 735

static VALUE
math_ldexp(VALUE unused_obj, VALUE x, VALUE n)
{
    return DBL2NUM(ldexp(Get_Double(x), NUM2INT(n)));
}

.lgamma(x) ⇒ Array, ...

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]

Returns ].

Returns:



902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
 
# File 'math.c', line 902

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(HUGE_VAL), INT2FIX(1));
    }
    if (d == 0.0) {
	VALUE vsign = signbit(d) ? INT2FIX(-1) : INT2FIX(+1);
	return rb_assoc_new(DBL2NUM(HUGE_VAL), vsign);
    }
    v = DBL2NUM(lgamma_r(d, &sign));
    return rb_assoc_new(v, INT2FIX(sign));
}

.log(x) ⇒ Float .log(x, base) ⇒ Float

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

Overloads:



452
453
454
455
456
 
# File 'math.c', line 452

static VALUE
math_log(int argc, const VALUE *argv, VALUE unused_obj)
{
    return rb_math_log(argc, argv);
}

.log10(x) ⇒ Float

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

Returns:



562
563
564
565
566
567
568
569
570
571
572
573
574
 
# File 'math.c', line 562

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(-HUGE_VAL);

    return DBL2NUM(log10(d) + numbits * log10(2)); /* log10(d * 2 ** numbits) */
}

.log2(x) ⇒ Float

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

Returns:



532
533
534
535
536
537
538
539
540
541
542
543
544
 
# File 'math.c', line 532

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(-HUGE_VAL);

    return DBL2NUM(log2(d) + numbits); /* log2(d * 2 ** numbits) */
}

.sin(x) ⇒ Float

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

Returns:



131
132
133
134
135
 
# File 'math.c', line 131

static VALUE
math_sin(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(sin(Get_Double(x)));
}

.sinh(x) ⇒ Float

Computes the hyperbolic sine of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.sinh(0) #=> 0.0

Returns:



276
277
278
279
280
 
# File 'math.c', line 276

static VALUE
math_sinh(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(sinh(Get_Double(x)));
}

.sqrt(x) ⇒ Float

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.

Returns:



611
612
613
614
615
 
# File 'math.c', line 611

static VALUE
math_sqrt(VALUE unused_obj, VALUE x)
{
    return rb_math_sqrt(x);
}

.tan(x) ⇒ Float

Computes the tangent of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.tan(0) #=> 0.0

Returns:



152
153
154
155
156
 
# File 'math.c', line 152

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

.tanh(x) ⇒ Float

Computes the hyperbolic tangent of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

Math.tanh(0) #=> 0.0

Returns:



311
312
313
314
315
 
# File 'math.c', line 311

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

Instance Method Details

#acos(x) ⇒ Float (private)

Computes the arc cosine of x. Returns 0..PI.

Domain: [-1, 1]

Codomain: [0, PI]

Math.acos(0) == Math::PI/2  #=> true

Returns:



172
173
174
175
176
177
178
179
180
181
 
# File 'math.c', line 172

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 (private)

Computes the inverse hyperbolic cosine of x.

Domain: [1, INFINITY)

Codomain: [0, INFINITY)

Math.acosh(1) #=> 0.0

Returns:



331
332
333
334
335
336
337
338
339
340
 
# File 'math.c', line 331

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 (private)

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

Returns:



196
197
198
199
200
201
202
203
204
205
 
# File 'math.c', line 196

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 (private)

Computes the inverse hyperbolic sine of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.asinh(1) #=> 0.881373587019543

Returns:



356
357
358
359
360
 
# File 'math.c', line 356

static VALUE
math_asinh(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(asinh(Get_Double(x)));
}

#atan(x) ⇒ Float (private)

Computes the arc tangent of x. Returns -PI/2..PI/2.

Domain: (-INFINITY, INFINITY)

Codomain: (-PI/2, PI/2)

Math.atan(0) #=> 0.0

Returns:



220
221
222
223
224
 
# File 'math.c', line 220

static VALUE
math_atan(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(atan(Get_Double(x)));
}

#atan2(y, x) ⇒ Float (private)

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

Returns:



65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
 
# File 'math.c', line 65

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 (private)

Computes the inverse hyperbolic tangent of x.

Domain: (-1, 1)

Codomain: (-INFINITY, INFINITY)

Math.atanh(1) #=> Infinity

Returns:



376
377
378
379
380
381
382
383
384
385
386
387
388
 
# File 'math.c', line 376

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(-HUGE_VAL);
    if (d == +1.0) return DBL2NUM(+HUGE_VAL);
    return DBL2NUM(atanh(d));
}

#cbrt(x) ⇒ Float (private)

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]

Returns:



691
692
693
694
695
696
697
698
699
700
701
702
 
# File 'math.c', line 691

static VALUE
math_cbrt(VALUE unused_obj, VALUE x)
{
    double f = Get_Double(x);
    double r = cbrt(f);
#if defined __GLIBC__
    if (isfinite(r)) {
	r = (2.0 * r + (f / r / r)) / 3.0;
    }
#endif
    return DBL2NUM(r);
}

#cos(x) ⇒ Float (private)

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

Returns:



110
111
112
113
114
 
# File 'math.c', line 110

static VALUE
math_cos(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(cos(Get_Double(x)));
}

#cosh(x) ⇒ Float (private)

Computes the hyperbolic cosine of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: [1, INFINITY)

Math.cosh(0) #=> 1.0

Returns:



248
249
250
251
252
 
# File 'math.c', line 248

static VALUE
math_cosh(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(cosh(Get_Double(x)));
}

#erf(x) ⇒ Float (private)

Calculates the error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

  Math.erf(0) #=> 0.0

Returns:



771
772
773
774
775
 
# File 'math.c', line 771

static VALUE
math_erf(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(erf(Get_Double(x)));
}

#erfc(x) ⇒ Float (private)

Calculates the complementary error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (0, 2)

  Math.erfc(0) #=> 1.0

Returns:



791
792
793
794
795
 
# File 'math.c', line 791

static VALUE
math_erfc(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(erfc(Get_Double(x)));
}

#exp(x) ⇒ Float (private)

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

Returns:



406
407
408
409
410
 
# File 'math.c', line 406

static VALUE
math_exp(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(exp(Get_Double(x)));
}

#frexp(x) ⇒ Array (private)

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

Returns:



715
716
717
718
719
720
721
722
723
 
# File 'math.c', line 715

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 (private)

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]

Returns:



837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
 
# File 'math.c', line 837

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(HUGE_VAL);
    }
    if (d == 0.0) {
	return signbit(d) ? DBL2NUM(-HUGE_VAL) : DBL2NUM(HUGE_VAL);
    }
    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 (private)

Returns sqrt(x**2 + y**2), the hypotenuse of a right-angled triangle with sides x and y.

Math.hypot(3, 4)   #=> 5.0

Returns:



751
752
753
754
755
 
# File 'math.c', line 751

static VALUE
math_hypot(VALUE unused_obj, VALUE x, VALUE y)
{
    return DBL2NUM(hypot(Get_Double(x), Get_Double(y)));
}

#ldexp(fraction, exponent) ⇒ Float (private)

Returns the value of fraction*(2**exponent).

fraction, exponent = Math.frexp(1234)
Math.ldexp(fraction, exponent)   #=> 1234.0

Returns:



735
736
737
738
739
 
# File 'math.c', line 735

static VALUE
math_ldexp(VALUE unused_obj, VALUE x, VALUE n)
{
    return DBL2NUM(ldexp(Get_Double(x), NUM2INT(n)));
}

#lgamma(x) ⇒ Array, ... (private)

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]

Returns ].

Returns:



902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
 
# File 'math.c', line 902

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(HUGE_VAL), INT2FIX(1));
    }
    if (d == 0.0) {
	VALUE vsign = signbit(d) ? INT2FIX(-1) : INT2FIX(+1);
	return rb_assoc_new(DBL2NUM(HUGE_VAL), vsign);
    }
    v = DBL2NUM(lgamma_r(d, &sign));
    return rb_assoc_new(v, INT2FIX(sign));
}

#log(x) ⇒ Float (private) #log(x, base) ⇒ Float (private)

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

Overloads:



452
453
454
455
456
 
# File 'math.c', line 452

static VALUE
math_log(int argc, const VALUE *argv, VALUE unused_obj)
{
    return rb_math_log(argc, argv);
}

#log10(x) ⇒ Float (private)

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

Returns:



562
563
564
565
566
567
568
569
570
571
572
573
574
 
# File 'math.c', line 562

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(-HUGE_VAL);

    return DBL2NUM(log10(d) + numbits * log10(2)); /* log10(d * 2 ** numbits) */
}

#log2(x) ⇒ Float (private)

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

Returns:



532
533
534
535
536
537
538
539
540
541
542
543
544
 
# File 'math.c', line 532

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(-HUGE_VAL);

    return DBL2NUM(log2(d) + numbits); /* log2(d * 2 ** numbits) */
}

#sin(x) ⇒ Float (private)

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

Returns:



131
132
133
134
135
 
# File 'math.c', line 131

static VALUE
math_sin(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(sin(Get_Double(x)));
}

#sinh(x) ⇒ Float (private)

Computes the hyperbolic sine of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.sinh(0) #=> 0.0

Returns:



276
277
278
279
280
 
# File 'math.c', line 276

static VALUE
math_sinh(VALUE unused_obj, VALUE x)
{
    return DBL2NUM(sinh(Get_Double(x)));
}

#sqrt(x) ⇒ Float (private)

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.

Returns:



611
612
613
614
615
 
# File 'math.c', line 611

static VALUE
math_sqrt(VALUE unused_obj, VALUE x)
{
    return rb_math_sqrt(x);
}

#tan(x) ⇒ Float (private)

Computes the tangent of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.tan(0) #=> 0.0

Returns:



152
153
154
155
156
 
# File 'math.c', line 152

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

#tanh(x) ⇒ Float (private)

Computes the hyperbolic tangent of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

Math.tanh(0) #=> 0.0

Returns:



311
312
313
314
315
 
# File 'math.c', line 311

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