# Module: BigMath

Defined in:
lib/bigdecimal/math.rb,
bigdecimal.c

## Overview

– Contents:

``````sqrt(x, prec)
sin (x, prec)
cos (x, prec)
atan(x, prec)  Note: |x|<1, x=0.9999 may not converge.
PI  (prec)
E   (prec) == exp(1.0,prec)
``````

where:

``````x    ... BigDecimal number to be computed.
|x| must be small enough to get convergence.
prec ... Number of digits to be obtained.
``````

++

Provides mathematical functions.

Example:

``````require "bigdecimal"
require "bigdecimal/math"

include BigMath

a = BigDecimal((PI(100)/2).to_s)
puts sin(a,100) # -> 0.10000000000000000000......E1
``````

## Class Method Summary collapse

• Computes the arctangent of x to the specified number of digits of precision.

• Computes the cosine of x to the specified number of digits of precision.

• Computes e (the base of natural logarithms) to the specified number of digits of precision.

• BigMath.exp(x, prec).

• BigMath.log(x, prec).

• Computes the value of pi to the specified number of digits of precision.

• Computes the sine of x to the specified number of digits of precision.

• Computes the square root of x to the specified number of digits of precision.

## Class Method Details

### .atan(x, prec) ⇒ Object

Computes the arctangent of x to the specified number of digits of precision.

If x is NaN, returns NaN.

Raises:

• (ArgumentError)
 ``` 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145``` ```# File 'lib/bigdecimal/math.rb', line 119 def atan(x, prec) raise ArgumentError, "Zero or negative precision for atan" if prec <= 0 return BigDecimal("NaN") if x.nan? pi = PI(prec) x = -x if neg = x < 0 return pi.div(neg ? -2 : 2, prec) if x.infinite? return pi / (neg ? -4 : 4) if x.round(prec) == 1 x = BigDecimal("1").div(x, prec) if inv = x > 1 x = (-1 + sqrt(1 + x**2, prec))/x if dbl = x > 0.5 n = prec + BigDecimal.double_fig y = x d = y t = x r = BigDecimal("3") x2 = x.mult(x,n) while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0) m = BigDecimal.double_fig if m < BigDecimal.double_fig t = -t.mult(x2,n) d = t.div(r,m) y += d r += 2 end y *= 2 if dbl y = pi / 2 - y if inv y = -y if neg y end```

### .cos(x, prec) ⇒ Object

Computes the cosine of x to the specified number of digits of precision.

If x is infinite or NaN, returns NaN.

Raises:

• (ArgumentError)
 ``` 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114``` ```# File 'lib/bigdecimal/math.rb', line 83 def cos(x, prec) raise ArgumentError, "Zero or negative precision for cos" if prec <= 0 return BigDecimal("NaN") if x.infinite? || x.nan? n = prec + BigDecimal.double_fig one = BigDecimal("1") two = BigDecimal("2") x = -x if x < 0 if x > (twopi = two * BigMath.PI(prec)) if x > 30 x %= twopi else x -= twopi while x > twopi end end x1 = one x2 = x.mult(x,n) sign = 1 y = one d = y i = BigDecimal("0") z = one while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0) m = BigDecimal.double_fig if m < BigDecimal.double_fig sign = -sign x1 = x2.mult(x1,n) i += two z *= (i-one) * i d = sign * x1.div(z,m) y += d end y end```

### .E(prec) ⇒ Object

Computes e (the base of natural logarithms) to the specified number of digits of precision.

Raises:

• (ArgumentError)
 ``` 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204``` ```# File 'lib/bigdecimal/math.rb', line 188 def E(prec) raise ArgumentError, "Zero or negative precision for E" if prec <= 0 n = prec + BigDecimal.double_fig one = BigDecimal("1") y = one d = y z = one i = 0 while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0) m = BigDecimal.double_fig if m < BigDecimal.double_fig i += 1 z *= i d = one.div(z,m) y += d end y end```

### .exp ⇒ Object

BigMath.exp(x, prec)

Computes the value of e (the base of natural logarithms) raised to the power of x, to the specified number of digits of precision.

If x is infinity, returns Infinity.

If x is NaN, returns NaN.

 ``` 2620 2621 2622 2623 2624 2625 2626 2627 2628 2629 2630 2631 2632 2633 2634 2635 2636 2637 2638 2639 2640 2641 2642 2643 2644 2645 2646 2647 2648 2649 2650 2651 2652 2653 2654 2655 2656 2657 2658 2659 2660 2661 2662 2663 2664 2665 2666 2667 2668 2669 2670 2671 2672 2673 2674 2675 2676 2677 2678 2679 2680 2681 2682 2683 2684 2685 2686 2687 2688 2689 2690 2691 2692 2693 2694 2695 2696 2697 2698 2699 2700 2701 2702 2703 2704 2705 2706 2707 2708 2709 2710 2711 2712 2713 2714 2715 2716 2717 2718 2719 2720 2721 2722 2723 2724 2725 2726 2727 2728 2729 2730 2731 2732 2733 2734 2735 2736 2737 2738``` ```# File 'bigdecimal.c', line 2620 static VALUE BigMath_s_exp(VALUE klass, VALUE x, VALUE vprec) { ssize_t prec, n, i; Real* vx = NULL; VALUE one, d, x1, y, z; int negative = 0; int infinite = 0; int nan = 0; double flo; prec = NUM2SSIZET(vprec); if (prec <= 0) { rb_raise(rb_eArgError, "Zero or negative precision for exp"); } /* TODO: the following switch statement is almostly the same as one in the * BigDecimalCmp function. */ switch (TYPE(x)) { case T_DATA: if (!is_kind_of_BigDecimal(x)) break; vx = DATA_PTR(x); negative = VpGetSign(vx) < 0; infinite = VpIsPosInf(vx) || VpIsNegInf(vx); nan = VpIsNaN(vx); break; case T_FIXNUM: /* fall through */ case T_BIGNUM: vx = GetVpValue(x, 0); break; case T_FLOAT: flo = RFLOAT_VALUE(x); negative = flo < 0; infinite = isinf(flo); nan = isnan(flo); if (!infinite && !nan) { vx = GetVpValueWithPrec(x, DBL_DIG+1, 0); } break; case T_RATIONAL: vx = GetVpValueWithPrec(x, prec, 0); break; default: break; } if (infinite) { if (negative) { return ToValue(GetVpValueWithPrec(INT2NUM(0), prec, 1)); } else { Real* vy; vy = VpCreateRbObject(prec, "#0"); RB_GC_GUARD(vy->obj); VpSetInf(vy, VP_SIGN_POSITIVE_INFINITE); return ToValue(vy); } } else if (nan) { Real* vy; vy = VpCreateRbObject(prec, "#0"); RB_GC_GUARD(vy->obj); VpSetNaN(vy); return ToValue(vy); } else if (vx == NULL) { cannot_be_coerced_into_BigDecimal(rb_eArgError, x); } RB_GC_GUARD(vx->obj); n = prec + rmpd_double_figures(); negative = VpGetSign(vx) < 0; if (negative) { VpSetSign(vx, 1); } RB_GC_GUARD(one) = ToValue(VpCreateRbObject(1, "1")); RB_GC_GUARD(x1) = one; RB_GC_GUARD(y) = one; RB_GC_GUARD(d) = y; RB_GC_GUARD(z) = one; i = 0; while (!VpIsZero((Real*)DATA_PTR(d))) { VALUE argv[2]; SIGNED_VALUE const ey = VpExponent10(DATA_PTR(y)); SIGNED_VALUE const ed = VpExponent10(DATA_PTR(d)); ssize_t m = n - vabs(ey - ed); if (m <= 0) { break; } else if ((size_t)m < rmpd_double_figures()) { m = rmpd_double_figures(); } x1 = BigDecimal_mult2(x1, x, SSIZET2NUM(n)); ++i; z = BigDecimal_mult(z, SSIZET2NUM(i)); argv[0] = z; argv[1] = SSIZET2NUM(m); d = BigDecimal_div2(2, argv, x1); y = BigDecimal_add(y, d); } if (negative) { VALUE argv[2]; argv[0] = y; argv[1] = vprec; return BigDecimal_div2(2, argv, one); } else { vprec = SSIZET2NUM(prec - VpExponent10(DATA_PTR(y))); return BigDecimal_round(1, &vprec, y); } }```

### .log ⇒ Object

BigMath.log(x, prec)

Computes the natural logarithm of x to the specified number of digits of precision.

If x is zero or negative, raises Math::DomainError.

If x is positive infinity, returns Infinity.

If x is NaN, returns NaN.

 ``` 2752 2753 2754 2755 2756 2757 2758 2759 2760 2761 2762 2763 2764 2765 2766 2767 2768 2769 2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 2781 2782 2783 2784 2785 2786 2787 2788 2789 2790 2791 2792 2793 2794 2795 2796 2797 2798 2799 2800 2801 2802 2803 2804 2805 2806 2807 2808 2809 2810 2811 2812 2813 2814 2815 2816 2817 2818 2819 2820 2821 2822 2823 2824 2825 2826 2827 2828 2829 2830 2831 2832 2833 2834 2835 2836 2837 2838 2839 2840 2841 2842 2843 2844 2845 2846 2847 2848 2849 2850 2851 2852 2853 2854 2855 2856 2857 2858 2859 2860 2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 2886 2887 2888 2889 2890 2891 2892 2893 2894 2895 2896 2897 2898 2899 2900``` ```# File 'bigdecimal.c', line 2752 static VALUE BigMath_s_log(VALUE klass, VALUE x, VALUE vprec) { ssize_t prec, n, i; SIGNED_VALUE expo; Real* vx = NULL; VALUE argv[2], vn, one, two, w, x2, y, d; int zero = 0; int negative = 0; int infinite = 0; int nan = 0; double flo; long fix; if (!is_integer(vprec)) { rb_raise(rb_eArgError, "precision must be an Integer"); } prec = NUM2SSIZET(vprec); if (prec <= 0) { rb_raise(rb_eArgError, "Zero or negative precision for exp"); } /* TODO: the following switch statement is almostly the same as one in the * BigDecimalCmp function. */ switch (TYPE(x)) { case T_DATA: if (!is_kind_of_BigDecimal(x)) break; vx = DATA_PTR(x); zero = VpIsZero(vx); negative = VpGetSign(vx) < 0; infinite = VpIsPosInf(vx) || VpIsNegInf(vx); nan = VpIsNaN(vx); break; case T_FIXNUM: fix = FIX2LONG(x); zero = fix == 0; negative = fix < 0; goto get_vp_value; case T_BIGNUM: zero = RBIGNUM_ZERO_P(x); negative = RBIGNUM_NEGATIVE_P(x); get_vp_value: if (zero || negative) break; vx = GetVpValue(x, 0); break; case T_FLOAT: flo = RFLOAT_VALUE(x); zero = flo == 0; negative = flo < 0; infinite = isinf(flo); nan = isnan(flo); if (!zero && !negative && !infinite && !nan) { vx = GetVpValueWithPrec(x, DBL_DIG+1, 1); } break; case T_RATIONAL: zero = RRATIONAL_ZERO_P(x); negative = RRATIONAL_NEGATIVE_P(x); if (zero || negative) break; vx = GetVpValueWithPrec(x, prec, 1); break; case T_COMPLEX: rb_raise(rb_eMathDomainError, "Complex argument for BigMath.log"); default: break; } if (infinite && !negative) { Real* vy; vy = VpCreateRbObject(prec, "#0"); RB_GC_GUARD(vy->obj); VpSetInf(vy, VP_SIGN_POSITIVE_INFINITE); return ToValue(vy); } else if (nan) { Real* vy; vy = VpCreateRbObject(prec, "#0"); RB_GC_GUARD(vy->obj); VpSetNaN(vy); return ToValue(vy); } else if (zero || negative) { rb_raise(rb_eMathDomainError, "Zero or negative argument for log"); } else if (vx == NULL) { cannot_be_coerced_into_BigDecimal(rb_eArgError, x); } x = ToValue(vx); RB_GC_GUARD(one) = ToValue(VpCreateRbObject(1, "1")); RB_GC_GUARD(two) = ToValue(VpCreateRbObject(1, "2")); n = prec + rmpd_double_figures(); RB_GC_GUARD(vn) = SSIZET2NUM(n); expo = VpExponent10(vx); if (expo < 0 || expo >= 3) { char buf[16]; snprintf(buf, 16, "1E%"PRIdVALUE, -expo); x = BigDecimal_mult2(x, ToValue(VpCreateRbObject(1, buf)), vn); } else { expo = 0; } w = BigDecimal_sub(x, one); argv[0] = BigDecimal_add(x, one); argv[1] = vn; x = BigDecimal_div2(2, argv, w); RB_GC_GUARD(x2) = BigDecimal_mult2(x, x, vn); RB_GC_GUARD(y) = x; RB_GC_GUARD(d) = y; i = 1; while (!VpIsZero((Real*)DATA_PTR(d))) { SIGNED_VALUE const ey = VpExponent10(DATA_PTR(y)); SIGNED_VALUE const ed = VpExponent10(DATA_PTR(d)); ssize_t m = n - vabs(ey - ed); if (m <= 0) { break; } else if ((size_t)m < rmpd_double_figures()) { m = rmpd_double_figures(); } x = BigDecimal_mult2(x2, x, vn); i += 2; argv[0] = SSIZET2NUM(i); argv[1] = SSIZET2NUM(m); d = BigDecimal_div2(2, argv, x); y = BigDecimal_add(y, d); } y = BigDecimal_mult(y, two); if (expo != 0) { VALUE log10, vexpo, dy; log10 = BigMath_s_log(klass, INT2FIX(10), vprec); vexpo = ToValue(GetVpValue(SSIZET2NUM(expo), 1)); dy = BigDecimal_mult(log10, vexpo); y = BigDecimal_add(y, dy); } return y; }```

### .PI(prec) ⇒ Object

Computes the value of pi to the specified number of digits of precision.

Raises:

• (ArgumentError)
 ``` 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184``` ```# File 'lib/bigdecimal/math.rb', line 148 def PI(prec) raise ArgumentError, "Zero or negative argument for PI" if prec <= 0 n = prec + BigDecimal.double_fig zero = BigDecimal("0") one = BigDecimal("1") two = BigDecimal("2") m25 = BigDecimal("-0.04") m57121 = BigDecimal("-57121") pi = zero d = one k = one w = one t = BigDecimal("-80") while d.nonzero? && ((m = n - (pi.exponent - d.exponent).abs) > 0) m = BigDecimal.double_fig if m < BigDecimal.double_fig t = t*m25 d = t.div(k,m) k = k+two pi = pi + d end d = one k = one w = one t = BigDecimal("956") while d.nonzero? && ((m = n - (pi.exponent - d.exponent).abs) > 0) m = BigDecimal.double_fig if m < BigDecimal.double_fig t = t.div(m57121,n) d = t.div(k,m) pi = pi + d k = k+two end pi end```

### .sin(x, prec) ⇒ Object

Computes the sine of x to the specified number of digits of precision.

If x is infinite or NaN, returns NaN.

Raises:

• (ArgumentError)
 ``` 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78``` ```# File 'lib/bigdecimal/math.rb', line 47 def sin(x, prec) raise ArgumentError, "Zero or negative precision for sin" if prec <= 0 return BigDecimal("NaN") if x.infinite? || x.nan? n = prec + BigDecimal.double_fig one = BigDecimal("1") two = BigDecimal("2") x = -x if neg = x < 0 if x > (twopi = two * BigMath.PI(prec)) if x > 30 x %= twopi else x -= twopi while x > twopi end end x1 = x x2 = x.mult(x,n) sign = 1 y = x d = y i = one z = one while d.nonzero? && ((m = n - (y.exponent - d.exponent).abs) > 0) m = BigDecimal.double_fig if m < BigDecimal.double_fig sign = -sign x1 = x2.mult(x1,n) i += two z *= (i-one) * i d = sign * x1.div(z,m) y += d end neg ? -y : y end```

### .sqrt(x, prec) ⇒ Object

Computes the square root of x to the specified number of digits of precision.

BigDecimal.new('2').sqrt(16).to_s

``````-> "0.14142135623730950488016887242096975E1"
``````
 ``` 40 41 42``` ```# File 'lib/bigdecimal/math.rb', line 40 def sqrt(x,prec) x.sqrt(prec) end```