diff options
Diffstat (limited to 'vendor/bundle/ruby/3.4.0/gems/bigdecimal-4.1.2/lib/bigdecimal/math.rb')
| -rw-r--r-- | vendor/bundle/ruby/3.4.0/gems/bigdecimal-4.1.2/lib/bigdecimal/math.rb | 927 |
1 files changed, 927 insertions, 0 deletions
diff --git a/vendor/bundle/ruby/3.4.0/gems/bigdecimal-4.1.2/lib/bigdecimal/math.rb b/vendor/bundle/ruby/3.4.0/gems/bigdecimal-4.1.2/lib/bigdecimal/math.rb new file mode 100644 index 0000000..1b3357c --- /dev/null +++ b/vendor/bundle/ruby/3.4.0/gems/bigdecimal-4.1.2/lib/bigdecimal/math.rb @@ -0,0 +1,927 @@ +# frozen_string_literal: true +require 'bigdecimal' + +# +#-- +# Contents: +# sqrt(x, prec) +# cbrt(x, prec) +# hypot(x, y, prec) +# sin (x, prec) +# cos (x, prec) +# tan (x, prec) +# asin(x, prec) +# acos(x, prec) +# atan(x, prec) +# atan2(y, x, prec) +# sinh (x, prec) +# cosh (x, prec) +# tanh (x, prec) +# asinh(x, prec) +# acosh(x, prec) +# atanh(x, prec) +# log2 (x, prec) +# log10(x, prec) +# log1p(x, prec) +# expm1(x, prec) +# erf (x, prec) +# erfc(x, prec) +# gamma(x, prec) +# lgamma(x, prec) +# frexp(x) +# ldexp(x, exponent) +# PI (prec) +# E (prec) == exp(1.0,prec) +# +# where: +# x, y ... BigDecimal number to be computed. +# prec ... Number of digits to be obtained. +#++ +# +# Provides mathematical functions. +# +# Example: +# +# require "bigdecimal/math" +# +# include BigMath +# +# a = BigDecimal((PI(49)/2).to_s) +# puts sin(a,100) # => 0.9999999999...9999999986e0 +# +module BigMath + module_function + + # 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'), 32).to_s + # #=> "0.14142135623730950488016887242097e1" + # + def sqrt(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :sqrt) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :sqrt) + x.sqrt(prec) + end + + + # Returns [sign, reduced_x] where reduced_x is in -pi/2..pi/2 + # and satisfies sin(x) = sign * sin(reduced_x) + # If add_half_pi is true, adds pi/2 to x before reduction. + # Precision of pi is adjusted to ensure reduced_x has the required precision. + private_class_method def _sin_periodic_reduction(x, prec, add_half_pi: false) # :nodoc: + return [1, x] if -Math::PI/2 <= x && x <= Math::PI/2 && !add_half_pi + + mod_prec = prec + BigDecimal::Internal::EXTRA_PREC + pi_extra_prec = [x.exponent, 0].max + BigDecimal::Internal::EXTRA_PREC + while true + pi = PI(mod_prec + pi_extra_prec) + half_pi = pi / 2 + div, mod = (add_half_pi ? x + pi : x + half_pi).divmod(pi) + mod -= half_pi + if mod.zero? || mod_prec + mod.exponent <= 0 + # mod is too small to estimate required pi precision + mod_prec = mod_prec * 3 / 2 + BigDecimal::Internal::EXTRA_PREC + elsif mod_prec + mod.exponent < prec + # Estimate required precision of pi + mod_prec = prec - mod.exponent + BigDecimal::Internal::EXTRA_PREC + else + return [div % 2 == 0 ? 1 : -1, mod.mult(1, prec)] + end + end + end + + private_class_method def _sin_binary_splitting(x, prec) # :nodoc: + return x if x.zero? + x2 = x.mult(x, prec) + # Find k that satisfies x2**k / (2k+1)! < 10**(-prec) + log10 = Math.log(10) + logx = BigDecimal::Internal.float_log(x.abs) + step = (1..).bsearch { |k| Math.lgamma(2 * k + 1)[0] - 2 * k * logx > prec * log10 } + # Construct denominator sequence for binary splitting + # sin(x) = x*(1-x2/(2*3)*(1-x2/(4*5)*(1-x2/(6*7)*(1-x2/(8*9)*(1-...))))) + ds = (1..step).map {|i| -(2 * i) * (2 * i + 1) } + x.mult(1 + BigDecimal::Internal.taylor_sum_binary_splitting(x2, ds, prec), prec) + end + + private_class_method def _sin_around_zero(x, prec) # :nodoc: + # Divide x into several parts + # sin(x.xxxxxxxx...) = sin(x.xx + 0.00xx + 0.0000xxxx + ...) + # Calculate sin of each part and restore sin(0.xxxxxxxx...) using addition theorem. + sin = BigDecimal(0) + cos = BigDecimal(1) + n = 2 + while x != 0 do + partial_x = x.truncate(n) + x -= partial_x + s = _sin_binary_splitting(partial_x, prec) + c = (1 - s * s).sqrt(prec) + sin, cos = (sin * c).add(cos * s, prec), (cos * c).sub(sin * s, prec) + n *= 2 + end + sin.clamp(BigDecimal(-1), BigDecimal(1)) + end + + # call-seq: + # cbrt(decimal, numeric) -> BigDecimal + # + # Computes the cube root of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # BigMath.cbrt(BigDecimal('2'), 32).to_s + # #=> "0.12599210498948731647672106072782e1" + # + def cbrt(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :cbrt) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :cbrt) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal::Internal.infinity_computation_result * x.infinite? if x.infinite? + return BigDecimal(0) if x.zero? + + x = -x if neg = x < 0 + ex = x.exponent / 3 + x = x._decimal_shift(-3 * ex) + y = BigDecimal(Math.cbrt(x.to_f), 0) + BigDecimal::Internal.newton_loop(prec + BigDecimal::Internal::EXTRA_PREC) do |p| + y = (2 * y + x.div(y, p).div(y, p)).div(3, p) + end + y._decimal_shift(ex).mult(neg ? -1 : 1, prec) + end + + # call-seq: + # hypot(x, y, numeric) -> BigDecimal + # + # Returns sqrt(x**2 + y**2) to the specified number of digits of + # precision, +numeric+. + # + # BigMath.hypot(BigDecimal('1'), BigDecimal('2'), 32).to_s + # #=> "0.22360679774997896964091736687313e1" + # + def hypot(x, y, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :hypot) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :hypot) + y = BigDecimal::Internal.coerce_to_bigdecimal(y, prec, :hypot) + return BigDecimal::Internal.nan_computation_result if x.nan? || y.nan? + return BigDecimal::Internal.infinity_computation_result if x.infinite? || y.infinite? + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + sqrt(x.mult(x, prec2) + y.mult(y, prec2), prec) + end + + # 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, 32).to_s + # #=> "0.70710807985947359435812921837984e0" + # + def sin(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :sin) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :sin) + return BigDecimal::Internal.nan_computation_result if x.infinite? || x.nan? + n = prec + BigDecimal::Internal::EXTRA_PREC + sign, x = _sin_periodic_reduction(x, n) + _sin_around_zero(x, n).mult(sign, prec) + end + + # 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(16), 32).to_s + # #=> "-0.99999999999999999999999999999997e0" + # + def cos(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :cos) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :cos) + return BigDecimal::Internal.nan_computation_result if x.infinite? || x.nan? + n = prec + BigDecimal::Internal::EXTRA_PREC + sign, x = _sin_periodic_reduction(x, n, add_half_pi: true) + _sin_around_zero(x, n).mult(sign, prec) + end + + # call-seq: + # tan(decimal, numeric) -> BigDecimal + # + # Computes the tangent of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is Infinity or NaN, returns NaN. + # + # BigMath.tan(BigDecimal("0.0"), 4).to_s + # #=> "0.0" + # + # BigMath.tan(BigMath.PI(24) / 4, 32).to_s + # #=> "0.99999999999999999999999830836025e0" + # + def tan(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :tan) + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + sin(x, prec2).div(cos(x, prec2), prec) + end + + # call-seq: + # asin(decimal, numeric) -> BigDecimal + # + # Computes the arcsine of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.asin(BigDecimal('0.5'), 32).to_s + # #=> "0.52359877559829887307710723054658e0" + # + def asin(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :asin) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :asin) + raise Math::DomainError, "Out of domain argument for asin" if x < -1 || x > 1 + return BigDecimal::Internal.nan_computation_result if x.nan? + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + cos = (1 - x**2).sqrt(prec2) + if cos.zero? + PI(prec2).div(x > 0 ? 2 : -2, prec) + else + atan(x.div(cos, prec2), prec) + end + end + + # call-seq: + # acos(decimal, numeric) -> BigDecimal + # + # Computes the arccosine of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.acos(BigDecimal('0.5'), 32).to_s + # #=> "0.10471975511965977461542144610932e1" + # + def acos(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :acos) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :acos) + raise Math::DomainError, "Out of domain argument for acos" if x < -1 || x > 1 + return BigDecimal::Internal.nan_computation_result if x.nan? + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + return (PI(prec2) / 2).sub(asin(x, prec2), prec) if x < 0 + return PI(prec2).div(2, prec) if x.zero? + + sin = (1 - x**2).sqrt(prec2) + atan(sin.div(x, prec2), prec) + end + + # 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'), 32).to_s + # #=> "-0.78539816339744830961566084581988e0" + # + def atan(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :atan) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :atan) + return BigDecimal::Internal.nan_computation_result if x.nan? + n = prec + BigDecimal::Internal::EXTRA_PREC + return PI(n).div(2 * x.infinite?, prec) if x.infinite? + + x = -x if neg = x < 0 + x = BigDecimal(1).div(x, n) if inv = x < -1 || x > 1 + + # Solve tan(y) - x = 0 with Newton's method + # Repeat: y -= (tan(y) - x) * cos(y)**2 + y = BigDecimal(Math.atan(x.to_f), 0) + BigDecimal::Internal.newton_loop(n) do |p| + s = sin(y, p) + c = (1 - s * s).sqrt(p) + y = y.sub(c * (s.sub(c * x.mult(1, p), p)), p) + end + y = PI(n) / 2 - y if inv + y.mult(neg ? -1 : 1, prec) + end + + # call-seq: + # atan2(decimal, decimal, numeric) -> BigDecimal + # + # Computes the arctangent of y and x to the specified number of digits of + # precision, +numeric+. + # + # BigMath.atan2(BigDecimal('-1'), BigDecimal('1'), 32).to_s + # #=> "-0.78539816339744830961566084581988e0" + # + def atan2(y, x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :atan2) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :atan2) + y = BigDecimal::Internal.coerce_to_bigdecimal(y, prec, :atan2) + return BigDecimal::Internal.nan_computation_result if x.nan? || y.nan? + + if x.infinite? || y.infinite? + one = BigDecimal(1) + zero = BigDecimal(0) + x = x.infinite? ? (x > 0 ? one : -one) : zero + y = y.infinite? ? (y > 0 ? one : -one) : y.sign * zero + end + + return x.sign >= 0 ? BigDecimal(0) : y.sign * PI(prec) if y.zero? + + y = -y if neg = y < 0 + xlarge = y.abs < x.abs + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + if x > 0 + v = xlarge ? atan(y.div(x, prec2), prec) : PI(prec2) / 2 - atan(x.div(y, prec2), prec2) + else + v = xlarge ? PI(prec2) - atan(-y.div(x, prec2), prec2) : PI(prec2) / 2 + atan(x.div(-y, prec2), prec2) + end + v.mult(neg ? -1 : 1, prec) + end + + # call-seq: + # sinh(decimal, numeric) -> BigDecimal + # + # Computes the hyperbolic sine of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.sinh(BigDecimal('1'), 32).to_s + # #=> "0.11752011936438014568823818505956e1" + # + def sinh(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :sinh) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :sinh) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal::Internal.infinity_computation_result * x.infinite? if x.infinite? + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + prec2 -= x.exponent if x.exponent < 0 + e = exp(x, prec2) + (e - BigDecimal(1).div(e, prec2)).div(2, prec) + end + + # call-seq: + # cosh(decimal, numeric) -> BigDecimal + # + # Computes the hyperbolic cosine of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.cosh(BigDecimal('1'), 32).to_s + # #=> "0.15430806348152437784779056207571e1" + # + def cosh(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :cosh) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :cosh) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal::Internal.infinity_computation_result if x.infinite? + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + e = exp(x, prec2) + (e + BigDecimal(1).div(e, prec2)).div(2, prec) + end + + # call-seq: + # tanh(decimal, numeric) -> BigDecimal + # + # Computes the hyperbolic tangent of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.tanh(BigDecimal('1'), 32).to_s + # #=> "0.76159415595576488811945828260479e0" + # + def tanh(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :tanh) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :tanh) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal(x.infinite?) if x.infinite? + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + [-x.exponent, 0].max + e = exp(x, prec2) + einv = BigDecimal(1).div(e, prec2) + (e - einv).div(e + einv, prec) + end + + # call-seq: + # asinh(decimal, numeric) -> BigDecimal + # + # Computes the inverse hyperbolic sine of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.asinh(BigDecimal('1'), 32).to_s + # #=> "0.88137358701954302523260932497979e0" + # + def asinh(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :asinh) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :asinh) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal::Internal.infinity_computation_result * x.infinite? if x.infinite? + return -asinh(-x, prec) if x < 0 + + sqrt_prec = prec + [-x.exponent, 0].max + BigDecimal::Internal::EXTRA_PREC + log(x + sqrt(x**2 + 1, sqrt_prec), prec) + end + + # call-seq: + # acosh(decimal, numeric) -> BigDecimal + # + # Computes the inverse hyperbolic cosine of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.acosh(BigDecimal('2'), 32).to_s + # #=> "0.1316957896924816708625046347308e1" + # + def acosh(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :acosh) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :acosh) + raise Math::DomainError, "Out of domain argument for acosh" if x < 1 + return BigDecimal::Internal.infinity_computation_result if x.infinite? + return BigDecimal::Internal.nan_computation_result if x.nan? + + log(x + sqrt(x**2 - 1, prec + BigDecimal::Internal::EXTRA_PREC), prec) + end + + # call-seq: + # atanh(decimal, numeric) -> BigDecimal + # + # Computes the inverse hyperbolic tangent of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.atanh(BigDecimal('0.5'), 32).to_s + # #=> "0.54930614433405484569762261846126e0" + # + def atanh(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :atanh) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :atanh) + raise Math::DomainError, "Out of domain argument for atanh" if x < -1 || x > 1 + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal::Internal.infinity_computation_result if x == 1 + return -BigDecimal::Internal.infinity_computation_result if x == -1 + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + (log(x + 1, prec2) - log(1 - x, prec2)).div(2, prec) + end + + # call-seq: + # BigMath.log2(decimal, numeric) -> BigDecimal + # + # Computes the base 2 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. + # + # BigMath.log2(BigDecimal('3'), 32).to_s + # #=> "0.15849625007211561814537389439478e1" + # + def log2(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :log2) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :log2) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal::Internal.infinity_computation_result if x.infinite? == 1 + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC * 3 / 2 + v = log(x, prec2).div(log(BigDecimal(2), prec2), prec2) + # Perform half-up rounding to calculate log2(2**n)==n correctly in every rounding mode + v = v.round(prec + BigDecimal::Internal::EXTRA_PREC - (v.exponent < 0 ? v.exponent : 0), BigDecimal::ROUND_HALF_UP) + v.mult(1, prec) + end + + # call-seq: + # BigMath.log10(decimal, numeric) -> BigDecimal + # + # Computes the base 10 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. + # + # BigMath.log10(BigDecimal('3'), 32).to_s + # #=> "0.47712125471966243729502790325512e0" + # + def log10(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :log10) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :log10) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal::Internal.infinity_computation_result if x.infinite? == 1 + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC * 3 / 2 + v = log(x, prec2).div(log(BigDecimal(10), prec2), prec2) + # Perform half-up rounding to calculate log10(10**n)==n correctly in every rounding mode + v = v.round(prec + BigDecimal::Internal::EXTRA_PREC - (v.exponent < 0 ? v.exponent : 0), BigDecimal::ROUND_HALF_UP) + v.mult(1, prec) + end + + # call-seq: + # BigMath.log1p(decimal, numeric) -> BigDecimal + # + # Computes log(1 + decimal) to the specified number of digits of precision, +numeric+. + # + # BigMath.log1p(BigDecimal('0.1'), 32).to_s + # #=> "0.95310179804324860043952123280765e-1" + # + def log1p(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :log1p) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :log1p) + raise Math::DomainError, 'Out of domain argument for log1p' if x < -1 + + return log(x + 1, prec) + end + + # call-seq: + # BigMath.expm1(decimal, numeric) -> BigDecimal + # + # Computes exp(decimal) - 1 to the specified number of digits of precision, +numeric+. + # + # BigMath.expm1(BigDecimal('0.1'), 32).to_s + # #=> "0.10517091807564762481170782649025e0" + # + def expm1(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :expm1) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :expm1) + return BigDecimal(-1) if x.infinite? == -1 + + exp_prec = prec + if x < -1 + # log10(exp(x)) = x * log10(e) + lg_e = 0.4342944819032518 + exp_prec = prec + (lg_e * x).ceil + BigDecimal::Internal::EXTRA_PREC + elsif x < 1 + exp_prec = prec - x.exponent + BigDecimal::Internal::EXTRA_PREC + else + exp_prec = prec + end + + return BigDecimal(-1) if exp_prec <= 0 + + exp(x, exp_prec).sub(1, prec) + end + + # call-seq: + # erf(decimal, numeric) -> BigDecimal + # + # Computes the error function of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.erf(BigDecimal('1'), 32).to_s + # #=> "0.84270079294971486934122063508261e0" + # + def erf(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :erf) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :erf) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal(x.infinite?) if x.infinite? + return BigDecimal(0) if x == 0 + return -erf(-x, prec) if x < 0 + return BigDecimal(1) if x > 5000000000 # erf(5000000000) > 1 - 1e-10000000000000000000 + + if x > 8 + xf = x.to_f + log10_erfc = -xf ** 2 / Math.log(10) - Math.log10(xf * Math::PI ** 0.5) + erfc_prec = [prec + log10_erfc.ceil, 1].max + erfc = _erfc_asymptotic(x, erfc_prec) + return BigDecimal(1).sub(erfc, prec) if erfc + end + + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + x_smallprec = x.mult(1, Integer.sqrt(prec2) / 2) + # Taylor series of x with small precision is fast + erf1 = _erf_taylor(x_smallprec, BigDecimal(0), BigDecimal(0), prec2) + # Taylor series converges quickly for small x + _erf_taylor(x - x_smallprec, x_smallprec, erf1, prec2).mult(1, prec) + end + + # call-seq: + # erfc(decimal, numeric) -> BigDecimal + # + # Computes the complementary error function of +decimal+ to the specified number of digits of + # precision, +numeric+. + # + # If +decimal+ is NaN, returns NaN. + # + # BigMath.erfc(BigDecimal('10'), 32).to_s + # #=> "0.20884875837625447570007862949578e-44" + # + def erfc(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :erfc) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :erfc) + return BigDecimal::Internal.nan_computation_result if x.nan? + return BigDecimal(1 - x.infinite?) if x.infinite? + return BigDecimal(1).sub(erf(x, prec + BigDecimal::Internal::EXTRA_PREC), prec) if x < 0.5 + return BigDecimal::Internal.underflow_computation_result if x > 5000000000 # erfc(5000000000) < 1e-10000000000000000000 (underflow) + + if x >= 8 + y = _erfc_asymptotic(x, prec) + return y.mult(1, prec) if y + end + + # erfc(x) = 1 - erf(x) < exp(-x**2)/x/sqrt(pi) + # Precision of erf(x) needs about log10(exp(-x**2)/x/sqrt(pi)) extra digits + log10 = 2.302585092994046 + xf = x.to_f + high_prec = prec + BigDecimal::Internal::EXTRA_PREC + ((xf**2 + Math.log(xf) + Math.log(Math::PI)/2) / log10).ceil + BigDecimal(1).sub(erf(x, high_prec), prec) + end + + # Calculates erf(x + a) + private_class_method def _erf_taylor(x, a, erf_a, prec) # :nodoc: + return erf_a if x.zero? + # Let f(x+a) = erf(x+a)*exp((x+a)**2)*sqrt(pi)/2 + # = c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + ... + # f'(x+a) = 1+2*(x+a)*f(x+a) + # f'(x+a) = c1 + 2*c2*x + 3*c3*x**2 + 4*c4*x**3 + 5*c5*x**4 + ... + # = 1+2*(x+a)*(c0 + c1*x + c2*x**2 + c3*x**3 + c4*x**4 + ...) + # therefore, + # c0 = f(a) + # c1 = 2 * a * c0 + 1 + # c2 = (2 * c0 + 2 * a * c1) / 2 + # c3 = (2 * c1 + 2 * a * c2) / 3 + # c4 = (2 * c2 + 2 * a * c3) / 4 + # + # All coefficients are positive when a >= 0 + + scale = BigDecimal(2).div(sqrt(PI(prec), prec), prec) + c_prev = erf_a.div(scale.mult(exp(-a*a, prec), prec), prec) + c_next = (2 * a * c_prev).add(1, prec).mult(x, prec) + sum = c_prev.add(c_next, prec) + + 2.step do |k| + cn = (c_prev.mult(x, prec) + a * c_next).mult(2, prec).mult(x, prec).div(k, prec) + sum = sum.add(cn, prec) + c_prev, c_next = c_next, cn + break if [c_prev, c_next].all? { |c| c.zero? || (c.exponent < sum.exponent - prec) } + end + value = sum.mult(scale.mult(exp(-(x + a).mult(x + a, prec), prec), prec), prec) + value > 1 ? BigDecimal(1) : value + end + + private_class_method def _erfc_asymptotic(x, prec) # :nodoc: + # Let f(x) = erfc(x)*sqrt(pi)*exp(x**2)/2 + # f(x) satisfies the following differential equation: + # 2*x*f(x) = f'(x) + 1 + # From the above equation, we can derive the following asymptotic expansion: + # f(x) = (0..kmax).sum { (-1)**k * (2*k)! / 4**k / k! / x**(2*k)) } / x + + # This asymptotic expansion does not converge. + # But if there is a k that satisfies (2*k)! / 4**k / k! / x**(2*k) < 10**(-prec), + # It is enough to calculate erfc within the given precision. + # Using Stirling's approximation, we can simplify this condition to: + # sqrt(2)/2 + k*log(k) - k - 2*k*log(x) < -prec*log(10) + # and the left side is minimized when k = x**2. + prec += BigDecimal::Internal::EXTRA_PREC + xf = x.to_f + kmax = (1..(xf ** 2).floor).bsearch do |k| + Math.log(2) / 2 + k * Math.log(k) - k - 2 * k * Math.log(xf) < -prec * Math.log(10) + end + return unless kmax + + sum = BigDecimal(1) + # To calculate `exp(x2, prec)`, x2 needs extra log10(x**2) digits of precision + x2 = x.mult(x, prec + (2 * Math.log10(xf)).ceil) + d = BigDecimal(1) + (1..kmax).each do |k| + d = d.div(x2, prec).mult(1 - 2 * k, prec).div(2, prec) + sum = sum.add(d, prec) + end + sum.div(exp(x2, prec).mult(PI(prec).sqrt(prec), prec), prec).div(x, prec) + end + + # call-seq: + # BigMath.gamma(decimal, numeric) -> BigDecimal + # + # Computes the gamma function of +decimal+ to the specified number of + # digits of precision, +numeric+. + # + # BigMath.gamma(BigDecimal('0.5'), 32).to_s + # #=> "0.17724538509055160272981674833411e1" + # + def gamma(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :gamma) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :gamma) + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + if x < 0.5 + raise Math::DomainError, 'Numerical argument is out of domain - gamma' if x.frac.zero? + + # Euler's reflection formula: gamma(z) * gamma(1-z) = pi/sin(pi*z) + pi = PI(prec2) + sin = _sinpix(x, pi, prec2) + return pi.div(gamma(1 - x, prec2).mult(sin, prec2), prec) + elsif x.frac.zero? && x < 1000 * prec + return _gamma_positive_integer(x, prec2).mult(1, prec) + end + + a, sum = _gamma_spouge_sum_part(x, prec2) + (x + (a - 1)).power(x - 0.5, prec2).mult(BigMath.exp(1 - x, prec2), prec2).mult(sum, prec) + end + + # call-seq: + # BigMath.lgamma(decimal, numeric) -> [BigDecimal, Integer] + # + # Computes the natural logarithm of the absolute value of the gamma function + # of +decimal+ to the specified number of digits of precision, +numeric+ and its sign. + # + # BigMath.lgamma(BigDecimal('0.5'), 32) + # #=> [0.57236494292470008707171367567653e0, 1] + # + def lgamma(x, prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :lgamma) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, prec, :lgamma) + prec2 = prec + BigDecimal::Internal::EXTRA_PREC + if x < 0.5 + return [BigDecimal::INFINITY, 1] if x.frac.zero? + + loop do + # Euler's reflection formula: gamma(z) * gamma(1-z) = pi/sin(pi*z) + pi = PI(prec2) + sin = _sinpix(x, pi, prec2) + log_gamma = BigMath.log(pi, prec2).sub(lgamma(1 - x, prec2).first + BigMath.log(sin.abs, prec2), prec) + return [log_gamma, sin > 0 ? 1 : -1] if prec2 + log_gamma.exponent > prec + BigDecimal::Internal::EXTRA_PREC + + # Retry with higher precision if loss of significance is too large + prec2 = prec2 * 3 / 2 + end + elsif x.frac.zero? && x < 1000 * prec + log_gamma = BigMath.log(_gamma_positive_integer(x, prec2), prec) + [log_gamma, 1] + else + # if x is close to 1 or 2, increase precision to reduce loss of significance + diff1_exponent = (x - 1).exponent + diff2_exponent = (x - 2).exponent + extremely_near_one = diff1_exponent < -prec2 + extremely_near_two = diff2_exponent < -prec2 + + if extremely_near_one || extremely_near_two + # If x is extreamely close to base = 1 or 2, linear interpolation is accurate enough. + # Taylor expansion at x = base is: (x - base) * digamma(base) + (x - base) ** 2 * trigamma(base) / 2 + ... + # And we can ignore (x - base) ** 2 and higher order terms. + base = extremely_near_one ? 1 : 2 + d = BigDecimal(1)._decimal_shift(1 - prec2) + log_gamma_d, sign = lgamma(base + d, prec2) + return [log_gamma_d.mult(x - base, prec2).div(d, prec), sign] + end + + prec2 += [-diff1_exponent, -diff2_exponent, 0].max + a, sum = _gamma_spouge_sum_part(x, prec2) + log_gamma = BigMath.log(sum, prec2).add((x - 0.5).mult(BigMath.log(x.add(a - 1, prec2), prec2), prec2) + 1 - x, prec) + [log_gamma, 1] + end + end + + # Returns sum part: sqrt(2*pi) and c[k]/(x+k) terms of Spouge's approximation + private_class_method def _gamma_spouge_sum_part(x, prec) # :nodoc: + x -= 1 + # Spouge's approximation + # x! = (x + a)**(x + 0.5) * exp(-x - a) * (sqrt(2 * pi) + (1..a - 1).sum{|k| c[k] / (x + k) } + epsilon) + # where c[k] = (-1)**k * (a - k)**(k - 0.5) * exp(a - k) / (k - 1)! + # and epsilon is bounded by a**(-0.5) * (2 * pi) ** (-a - 0.5) + + # Estimate required a for given precision + a = (prec / Math.log10(2 * Math::PI)).ceil + + # Calculate exponent of c[k] in low precision to estimate required precision + low_prec = 16 + log10f = Math.log(10) + x_low_prec = x.mult(1, low_prec) + loggamma_k = 0 + ck_exponents = (1..a-1).map do |k| + loggamma_k += Math.log10(k - 1) if k > 1 + -loggamma_k - k / log10f + (k - 0.5) * Math.log10(a - k) - BigDecimal::Internal.float_log(x_low_prec.add(k, low_prec)) / log10f + end + + # Estimate exponent of sum by Stirling's approximation + approx_sum_exponent = x < 1 ? -Math.log10(a) / 2 : Math.log10(2 * Math::PI) / 2 + x_low_prec.add(0.5, low_prec) * Math.log10(x_low_prec / x_low_prec.add(a, low_prec)) + + # Determine required precision of c[k] + prec2 = [ck_exponents.max.ceil - approx_sum_exponent.floor, 0].max + prec + + einv = BigMath.exp(-1, prec2) + sum = (PI(prec) * 2).sqrt(prec).mult(BigMath.exp(-a, prec), prec) + y = BigDecimal(1) + (1..a - 1).each do |k| + # c[k] = (-1)**k * (a - k)**(k - 0.5) * exp(-k) / (k-1)! / (x + k) + y = y.div(1 - k, prec2) if k > 1 + y = y.mult(einv, prec2) + z = y.mult(BigDecimal((a - k) ** k), prec2).div(BigDecimal(a - k).sqrt(prec2).mult(x.add(k, prec2), prec2), prec2) + # sum += c[k] / (x + k) + sum = sum.add(z, prec2) + end + [a, sum] + end + + private_class_method def _gamma_positive_integer(x, prec) # :nodoc: + return x if x == 1 + numbers = (1..x - 1).map {|i| BigDecimal(i) } + while numbers.size > 1 + numbers = numbers.each_slice(2).map {|a, b| b ? a.mult(b, prec) : a } + end + numbers.first + end + + # Returns sin(pi * x), for gamma reflection formula calculation + private_class_method def _sinpix(x, pi, prec) # :nodoc: + x = x % 2 + sign = x > 1 ? -1 : 1 + x %= 1 + x = 1 - x if x > 0.5 # to avoid sin(pi*x) loss of precision for x close to 1 + sign * sin(x.mult(pi, prec), prec) + end + + # call-seq: + # frexp(x) -> [BigDecimal, Integer] + # + # Decomposes +x+ into a normalized fraction and an integral power of ten. + # + # BigMath.frexp(BigDecimal(123.456)) + # #=> [0.123456e0, 3] + # + def frexp(x) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, 0, :frexp) + return [x, 0] unless x.finite? + + exponent = x.exponent + [x._decimal_shift(-exponent), exponent] + end + + # call-seq: + # ldexp(fraction, exponent) -> BigDecimal + # + # Inverse of +frexp+. + # Returns the value of fraction * 10**exponent. + # + # BigMath.ldexp(BigDecimal("0.123456e0"), 3) + # #=> 0.123456e3 + # + def ldexp(x, exponent) + x = BigDecimal::Internal.coerce_to_bigdecimal(x, 0, :ldexp) + x.finite? ? x._decimal_shift(exponent) : x + end + + # call-seq: + # PI(numeric) -> BigDecimal + # + # Computes the value of pi to the specified number of digits of precision, + # +numeric+. + # + # BigMath.PI(32).to_s + # #=> "0.31415926535897932384626433832795e1" + # + def PI(prec) + # Gauss–Legendre algorithm + prec = BigDecimal::Internal.coerce_validate_prec(prec, :PI) + n = prec + BigDecimal::Internal::EXTRA_PREC + a = BigDecimal(1) + b = BigDecimal(0.5, 0).sqrt(n) + s = BigDecimal(0.25, 0) + t = 1 + while a != b && (a - b).exponent > 1 - n + c = (a - b).div(2, n) + a, b = (a + b).div(2, n), (a * b).sqrt(n) + s = s.sub(c * c * t, n) + t *= 2 + end + (a * b).div(s, prec) + end + + # call-seq: + # E(numeric) -> BigDecimal + # + # Computes e (the base of natural logarithms) to the specified number of + # digits of precision, +numeric+. + # + # BigMath.E(32).to_s + # #=> "0.27182818284590452353602874713527e1" + # + def E(prec) + prec = BigDecimal::Internal.coerce_validate_prec(prec, :E) + exp(1, prec) + end +end |