Merge pull request rawify#22 from harrysarson/expm1

Implements expm1

Author: infusion

complex.js

@@ -45,6 +45,38 @@
     return (Math.exp(x) - Math.exp(-x)) * 0.5;
   };
 
+  /**
+   * Calculates cos(x) - 1 using Taylor series if x is small.
+   *
+   * @param {number} x
+   * @returns {number} cos(x) - 1
+   */
+
+  const cosm1 = function(x) {
+    const limit = Math.PI/4;
+    if (x < -limit || x > limit) {
+      return (Math.cos(x) - 1.0);
+    }
+
+    let xx = x * x;
+    return xx *
+      (-0.5 + xx *
+        (1/24 + xx *
+          (-1/720 + xx *
+            (1/40320 + xx *
+              (-1/3628800 + xx *
+                (1/4790014600 + xx *
+                  (-1/87178291200 + xx *
+                    (1/20922789888000)
+                  )
+                )
+              )
+            )
+          )
+        )
+      )
+  };
+
   const hypot = function(x, y) {
 
     let a = Math.abs(x);
@@ -525,6 +557,30 @@
               tmp * Math.sin(this['im']));
     },
 
+    /**
+     * Calculate the complex exponent and subtracts one.
+     *
+     * This may be more accurate than `Complex(x).exp().sub(1)` if
+     * `x` is small.
+     *
+     * @returns {Complex}
+     */
+    'expm1': function() {
+
+      /**
+       * exp(a + i*b) - 1
+       = exp(a) * (cos(b) + j*sin(b)) - 1
+       = expm1(a)*cos(b) + cosm1(b) + j*exp(a)*sin(b)
+       */
+
+      const a = this['re'];
+      const b = this['im'];
+
+      return new Complex(
+              Math.expm1(a) * Math.cos(b) + cosm1(b),
+              Math.exp(a) * Math.sin(b));
+    },
+
     /**
      * Calculate the natural log
      *

tests/complex.test.js

@@ -197,6 +197,26 @@ var functionTests = [{
     set: "3 - 2i",
     fn: "exp",
     expect: "-8.358532650935372 - 18.263727040666765i"
+  }, {
+    set: "3 - 2i",
+    fn: "expm1",
+    expect: "-9.358532650935372 - 18.263727040666765i"
+  }, {
+    set: "0",
+    fn: "expm1",
+    expect: "0"
+  }, {
+    set: "1e-6",
+    fn: "expm1",
+    expect: "0.0000010000005000001665"
+  }, {
+    set: "1e-5 + 5i",
+    fn: "expm1",
+    expect: "-0.716334977900736 - 0.9589338639538314i"
+  }, {
+    set: "1.2e-7 - 2e-6i",
+    fn: "expm1",
+    expect: "1.1999800719976027e-7 - 0.000002000000239998681i"
   }, {
     set: "3",
     fn: "pow",