# 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/math"

include BigMath

a = BigDecimal((PI(100)/2).to_s)
puts sin(a,100) # => 0.99999999999999999999......e0
``````

## Class Method Summary collapse

• call-seq: atan(decimal, numeric) -> BigDecimal.

• call-seq: cos(decimal, numeric) -> BigDecimal.

• call-seq: E(numeric) -> BigDecimal.

• BigMath.exp(decimal, numeric) -> BigDecimal.

• BigMath.log(decimal, numeric) -> BigDecimal.

• call-seq: PI(numeric) -> BigDecimal.

• call-seq: sin(decimal, numeric) -> BigDecimal.

• call-seq: sqrt(decimal, numeric) -> BigDecimal.

## Class Method Details

### .atan(x, prec) ⇒ Object

call-seq:

``````atan(decimal, numeric) -> BigDecimal
``````

Computes the arctangent of `decimal` to the specified number of digits of precision, `numeric`.

If `decimal` is NaN, returns NaN.

``````BigMath.atan(BigDecimal('-1'), 16).to_s
#=> "-0.785398163397448309615660845819878471907514682065e0"
``````

Raises:

• (ArgumentError)
 ``` 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172``` ```# File 'lib/bigdecimal/math.rb', line 146 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

call-seq:

``````cos(decimal, numeric) -> BigDecimal
``````

Computes the cosine of `decimal` to the specified number of digits of precision, `numeric`.

If `decimal` is Infinity or NaN, returns NaN.

``````BigMath.cos(BigMath.PI(4), 16).to_s
#=> "-0.999999999999999999999999999999856613163740061349e0"
``````

Raises:

• (ArgumentError)
 ``` 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133``` ```# File 'lib/bigdecimal/math.rb', line 102 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

call-seq:

``````E(numeric) -> BigDecimal
``````

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

``````BigMath.E(10).to_s
#=> "0.271828182845904523536028752390026306410273e1"
``````

Raises:

• (ArgumentError)
 ``` 228 229 230 231``` ```# File 'lib/bigdecimal/math.rb', line 228 def E(prec) raise ArgumentError, "Zero or negative precision for E" if prec <= 0 BigMath.exp(1, prec) end```

### .exp(x, vprec) ⇒ Object

BigMath.exp(decimal, numeric) -> BigDecimal

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

If `decimal` is infinity, returns Infinity.

If `decimal` is NaN, returns NaN.

 ``` 3048 3049 3050 3051 3052 3053 3054 3055 3056 3057 3058 3059 3060 3061 3062 3063 3064 3065 3066 3067 3068 3069 3070 3071 3072 3073 3074 3075 3076 3077 3078 3079 3080 3081 3082 3083 3084 3085 3086 3087 3088 3089 3090 3091 3092 3093 3094 3095 3096 3097 3098 3099 3100 3101 3102 3103 3104 3105 3106 3107 3108 3109 3110 3111 3112 3113 3114 3115 3116 3117 3118 3119 3120 3121 3122 3123 3124 3125 3126 3127 3128 3129 3130 3131 3132 3133 3134 3135 3136 3137 3138 3139 3140 3141 3142 3143 3144 3145 3146 3147 3148 3149 3150 3151 3152 3153 3154 3155 3156 3157 3158 3159 3160 3161 3162 3163 3164 3165 3166 3167 3168 3169``` ```# File 'bigdecimal.c', line 3048 static VALUE BigMath_s_exp(VALUE klass, VALUE x, VALUE vprec) { ssize_t prec, n, i; Real* vx = NULL; VALUE one, d, y; 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 almost 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 = BIGDECIMAL_NEGATIVE_P(vx); 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, DBLE_FIG, 0); } break; case T_RATIONAL: vx = GetVpValueWithPrec(x, prec, 0); break; default: break; } if (infinite) { if (negative) { return ToValue(GetVpValueWithPrec(INT2FIX(0), prec, 1)); } else { Real* vy; vy = VpCreateRbObject(prec, "#0"); VpSetInf(vy, VP_SIGN_POSITIVE_INFINITE); RB_GC_GUARD(vy->obj); return ToValue(vy); } } else if (nan) { Real* vy; vy = VpCreateRbObject(prec, "#0"); VpSetNaN(vy); RB_GC_GUARD(vy->obj); return ToValue(vy); } else if (vx == NULL) { cannot_be_coerced_into_BigDecimal(rb_eArgError, x); } x = vx->obj; n = prec + rmpd_double_figures(); negative = BIGDECIMAL_NEGATIVE_P(vx); if (negative) { VALUE x_zero = INT2NUM(1); VALUE x_copy = f_BigDecimal(1, &x_zero, klass); x = BigDecimal_initialize_copy(x_copy, x); vx = DATA_PTR(x); VpSetSign(vx, 1); } one = ToValue(VpCreateRbObject(1, "1")); y = one; 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); rb_thread_check_ints(); if (m <= 0) { break; } else if ((size_t)m < rmpd_double_figures()) { m = rmpd_double_figures(); } d = BigDecimal_mult(d, x); /* d <- d * x */ d = BigDecimal_div2(d, SSIZET2NUM(i), SSIZET2NUM(m)); /* d <- d / i */ y = BigDecimal_add(y, d); /* y <- y + d */ ++i; /* i <- i + 1 */ } if (negative) { return BigDecimal_div2(one, y, vprec); } else { vprec = SSIZET2NUM(prec - VpExponent10(DATA_PTR(y))); return BigDecimal_round(1, &vprec, y); } RB_GC_GUARD(one); RB_GC_GUARD(x); RB_GC_GUARD(y); RB_GC_GUARD(d); }```

### .log(x, vprec) ⇒ Object

BigMath.log(decimal, numeric) -> BigDecimal

Computes the natural logarithm of `decimal` to the specified number of digits of precision, `numeric`.

If `decimal` is zero or negative, raises Math::DomainError.

If `decimal` is positive infinity, returns Infinity.

If `decimal` is NaN, returns NaN.

 ``` 3183 3184 3185 3186 3187 3188 3189 3190 3191 3192 3193 3194 3195 3196 3197 3198 3199 3200 3201 3202 3203 3204 3205 3206 3207 3208 3209 3210 3211 3212 3213 3214 3215 3216 3217 3218 3219 3220 3221 3222 3223 3224 3225 3226 3227 3228 3229 3230 3231 3232 3233 3234 3235 3236 3237 3238 3239 3240 3241 3242 3243 3244 3245 3246 3247 3248 3249 3250 3251 3252 3253 3254 3255 3256 3257 3258 3259 3260 3261 3262 3263 3264 3265 3266 3267 3268 3269 3270 3271 3272 3273 3274 3275 3276 3277 3278 3279 3280 3281 3282 3283 3284 3285 3286 3287 3288 3289 3290 3291 3292 3293 3294 3295 3296 3297 3298 3299 3300 3301 3302 3303 3304 3305 3306 3307 3308 3309 3310 3311 3312 3313 3314 3315 3316 3317 3318 3319 3320 3321 3322 3323 3324 3325 3326 3327 3328``` ```# File 'bigdecimal.c', line 3183 static VALUE BigMath_s_log(VALUE klass, VALUE x, VALUE vprec) { ssize_t prec, n, i; SIGNED_VALUE expo; Real* vx = NULL; VALUE 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 almost 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 = BIGDECIMAL_NEGATIVE_P(vx); 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: i = FIX2INT(rb_big_cmp(x, INT2FIX(0))); zero = i == 0; negative = i < 0; 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, DBLE_FIG, 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[DECIMAL_SIZE_OF_BITS(SIZEOF_VALUE * CHAR_BIT) + 4]; snprintf(buf, sizeof(buf), "1E%"PRIdVALUE, -expo); x = BigDecimal_mult2(x, ToValue(VpCreateRbObject(1, buf)), vn); } else { expo = 0; } w = BigDecimal_sub(x, one); x = BigDecimal_div2(w, BigDecimal_add(x, one), vn); 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; d = BigDecimal_div2(x, SSIZET2NUM(i), SSIZET2NUM(m)); 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

call-seq:

``````PI(numeric) -> BigDecimal
``````

Computes the value of pi to the specified number of digits of precision, `numeric`.

``````BigMath.PI(10).to_s
#=> "0.3141592653589793238462643388813853786957412e1"
``````

Raises:

• (ArgumentError)
 ``` 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217``` ```# File 'lib/bigdecimal/math.rb', line 183 def PI(prec) raise ArgumentError, "Zero or negative precision 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 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 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

call-seq:

``````sin(decimal, numeric) -> BigDecimal
``````

Computes the sine of `decimal` to the specified number of digits of precision, `numeric`.

If `decimal` is Infinity or NaN, returns NaN.

``````BigMath.sin(BigMath.PI(5)/4, 5).to_s
#=> "0.70710678118654752440082036563292800375e0"
``````

Raises:

• (ArgumentError)
 ``` 58 59 60 61 62 63 64 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``` ```# File 'lib/bigdecimal/math.rb', line 58 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

call-seq:

``````sqrt(decimal, numeric) -> BigDecimal
``````

Computes the square root of `decimal` to the specified number of digits of precision, `numeric`.

``````BigMath.sqrt(BigDecimal('2'), 16).to_s
#=> "0.1414213562373095048801688724e1"
``````
 ``` 43 44 45``` ```# File 'lib/bigdecimal/math.rb', line 43 def sqrt(x, prec) x.sqrt(prec) end```