Faster exp calculation

lib/bigdecimal.rb

@@ -248,6 +248,18 @@ def self.log(x, prec)
     y.mult(2 ** (sqrt_steps + 1), prec)
   end
 
+  private_class_method def self._exp_taylor(x, prec) # :nodoc:
+    xn = BigDecimal(1)
+    y = BigDecimal(1)
+    1.step do |i|
+      n = prec + xn.exponent
+      break if n <= 0 || xn.zero?
+      xn = xn.mult(x, n).div(i, n)
+      y = y.add(xn, prec)
+    end
+    y
+  end
+
   # call-seq:
   #   BigMath.exp(decimal, numeric)    -> BigDecimal
   #
@@ -270,17 +282,12 @@ def self.exp(x, prec)
     cnt = x > 1 ? x.exponent : 0
     prec2 = prec + BigDecimal.double_fig + cnt
     x *= BigDecimal("1e-#{cnt}")
-    xn = BigDecimal(1)
-    y = BigDecimal(1)
 
-    # Taylor series for exp(x) around 0
-    1.step do |i|
-      n = prec2 + xn.exponent
-      break if n <= 0 || xn.zero?
-      x = x.mult(1, n)
-      xn = xn.mult(x, n).div(i, n)
-      y = y.add(xn, prec2)
-    end
+    # Calculation of exp(small_prec) is fast because calculation of x**n is fast
+    # Calculation of exp(small_abs) converges fast.
+    # exp(x) = exp(small_prec_part + small_abs_part) = exp(small_prec_part) * exp(small_abs_part)
+    x_small_prec = x.round(Integer.sqrt(prec2))
+    y = _exp_taylor(x_small_prec, prec2).mult(_exp_taylor(x - x_small_prec, prec2), prec2)
 
     # calculate exp(x * 10**cnt) from exp(x)
     # exp(x * 10**k) = exp(x * 10**(k - 1)) ** 10