Hi,
this adds the expm1 method to java.lang.StrictMath.
Please comment/commit.
Thanks,
Carsten
2006-07-22 Carsten Neumann <[EMAIL PROTECTED]>
* StrictMath.java (expm1): New method.
(EXPM1_Q1): New field.
(EXPM1_Q2): Likewise.
(EXPM1_Q3): Likewise.
(EXPM1_Q4): Likewise.
(EXPM1_Q6): Likewise.
Index: java/lang/StrictMath.java
===================================================================
RCS file: /sources/classpath/classpath/java/lang/StrictMath.java,v
retrieving revision 1.10
diff -u -r1.10 StrictMath.java
--- java/lang/StrictMath.java 16 Jul 2006 20:23:49 -0000 1.10
+++ java/lang/StrictMath.java 22 Jul 2006 14:50:39 -0000
@@ -819,6 +819,254 @@
}
/**
+ * Returns <em>e</em><sup>x</sup> - 1.
+ * Special cases:
+ * <ul>
+ * <li>If the argument is NaN, the result is NaN.</li>
+ * <li>If the argument is positive infinity, the result is positive
+ * infinity</li>
+ * <li>If the argument is negative infinity, the result is -1.</li>
+ * <li>If the argument is zero, the result is zero.</li>
+ * </ul>
+ *
+ * @param x the argument to <em>e</em><sup>x</sup> - 1.
+ * @return <em>e</em> raised to the power <code>x</code> minus one.
+ * @see #exp(double)
+ */
+ public static double expm1(double x)
+ {
+ // Method
+ // 1. Argument reduction:
+ // Given x, find r and integer k such that
+ //
+ // x = k * ln(2) + r, |r| <= 0.5 * ln(2)
+ //
+ // Here a correction term c will be computed to compensate
+ // the error in r when rounded to a floating-point number.
+ //
+ // 2. Approximating expm1(r) by a special rational function on
+ // the interval [0, 0.5 * ln(2)]:
+ // Since
+ // r*(exp(r)+1)/(exp(r)-1) = 2 + r^2/6 - r^4/360 + ...
+ // we define R1(r*r) by
+ // r*(exp(r)+1)/(exp(r)-1) = 2 + r^2/6 * R1(r*r)
+ // That is,
+ // R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
+ // = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
+ // = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
+ // We use a special Remes algorithm on [0, 0.347] to generate
+ // a polynomial of degree 5 in r*r to approximate R1. The
+ // maximum error of this polynomial approximation is bounded
+ // by 2**-61. In other words,
+ // R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
+ // where Q1 = -1.6666666666666567384E-2,
+ // Q2 = 3.9682539681370365873E-4,
+ // Q3 = -9.9206344733435987357E-6,
+ // Q4 = 2.5051361420808517002E-7,
+ // Q5 = -6.2843505682382617102E-9;
+ // (where z=r*r, and Q1 to Q5 are called EXPM1_Qx in the source)
+ // with error bounded by
+ // | 5 | -61
+ // | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
+ // | |
+ //
+ // expm1(r) = exp(r)-1 is then computed by the following
+ // specific way which minimize the accumulation rounding error:
+ // 2 3
+ // r r [ 3 - (R1 + R1*r/2) ]
+ // expm1(r) = r + --- + --- * [--------------------]
+ // 2 2 [ 6 - r*(3 - R1*r/2) ]
+ //
+ // To compensate the error in the argument reduction, we use
+ // expm1(r+c) = expm1(r) + c + expm1(r)*c
+ // ~ expm1(r) + c + r*c
+ // Thus c+r*c will be added in as the correction terms for
+ // expm1(r+c). Now rearrange the term to avoid optimization
+ // screw up:
+ // ( 2 2 )
+ // ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
+ // expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
+ // ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
+ // ( )
+ //
+ // = r - E
+ // 3. Scale back to obtain expm1(x):
+ // From step 1, we have
+ // expm1(x) = either 2^k*[expm1(r)+1] - 1
+ // = or 2^k*[expm1(r) + (1-2^-k)]
+ // 4. Implementation notes:
+ // (A). To save one multiplication, we scale the coefficient Qi
+ // to Qi*2^i, and replace z by (x^2)/2.
+ // (B). To achieve maximum accuracy, we compute expm1(x) by
+ // (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
+ // (ii) if k=0, return r-E
+ // (iii) if k=-1, return 0.5*(r-E)-0.5
+ // (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
+ // else return 1.0+2.0*(r-E);
+ // (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
+ // (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
+ // (vii) return 2^k(1-((E+2^-k)-r))
+
+ boolean negative = (x < 0);
+ double y, hi, lo, c, t, e, hxs, hfx, r1;
+ int k;
+
+ long bits;
+ int h_bits;
+ int l_bits;
+
+ c = 0.0;
+ y = abs(x);
+
+ bits = Double.doubleToLongBits(y);
+ h_bits = getHighDWord(bits);
+ l_bits = getLowDWord(bits);
+
+ // handle special cases and large arguments
+ if (h_bits >= 0x4043687a) // if |x| >= 56 * ln(2)
+ {
+ if (h_bits >= 0x40862e42) // if |x| >= EXP_LIMIT_H
+ {
+ if (h_bits >= 0x7ff00000)
+ {
+ if (((h_bits & 0x000fffff) | (l_bits & 0xffffffff)) != 0)
+ return Double.NaN; // exp(NaN) = NaN
+ else
+ return negative ? -1.0 : x; // exp({+-inf}) = {+inf, -1}
+ }
+
+ if (x > EXP_LIMIT_H)
+ return Double.POSITIVE_INFINITY; // overflow
+ }
+
+ if (negative) // x <= -56 * ln(2)
+ return -1.0;
+ }
+
+ // argument reduction
+ if (h_bits > 0x3fd62e42) // |x| > 0.5 * ln(2)
+ {
+ if (h_bits < 0x3ff0a2b2) // |x| < 1.5 * ln(2)
+ {
+ if (negative)
+ {
+ hi = x + LN2_H;
+ lo = -LN2_L;
+ k = -1;
+ }
+ else
+ {
+ hi = x - LN2_H;
+ lo = LN2_L;
+ k = 1;
+ }
+ }
+ else
+ {
+ k = (int) (INV_LN2 * x + (negative ? - 0.5 : 0.5));
+ t = k;
+ hi = x - t * LN2_H;
+ lo = t * LN2_L;
+ }
+
+ x = hi - lo;
+ c = (hi - x) - lo;
+
+ }
+ else if (h_bits < 0x3c900000) // |x| < 2^-54 return x
+ return x;
+ else
+ k = 0;
+
+ // x is now in primary range
+ hfx = 0.5 * x;
+ hxs = x * hfx;
+ r1 = 1.0 + hxs * (EXPM1_Q1
+ + hxs * (EXPM1_Q2
+ + hxs * (EXPM1_Q3
+ + hxs * (EXPM1_Q4
+ + hxs * EXPM1_Q5))));
+ t = 3.0 - r1 * hfx;
+ e = hxs * ((r1 - t) / (6.0 - x * t));
+
+ if (k == 0)
+ {
+ return x - (x * e - hxs); // c == 0
+ }
+ else
+ {
+ e = x * (e - c) - c;
+ e -= hxs;
+
+ if (k == -1)
+ return 0.5 * (x - e) - 0.5;
+
+ if (k == 1)
+ {
+ if (x < - 0.25)
+ return -2.0 * (e - (x + 0.5));
+ else
+ return 1.0 + 2.0 * (x - e);
+ }
+
+ if (k <= -2 || k > 56) // sufficient to return exp(x) - 1
+ {
+ y = 1.0 - (e - x);
+
+ bits = Double.doubleToLongBits(y);
+ h_bits = getHighDWord(bits);
+ l_bits = getLowDWord(bits);
+
+ h_bits += (k << 20); // add k to y's exponent
+
+ y = buildDouble(l_bits, h_bits);
+
+ return y - 1.0;
+ }
+
+ t = 1.0;
+ if (k < 20)
+ {
+ bits = Double.doubleToLongBits(t);
+ h_bits = 0x3ff00000 - (0x00200000 >> k);
+ l_bits = getLowDWord(bits);
+
+ t = buildDouble(l_bits, h_bits); // t = 1 - 2^(-k)
+ y = t - (e - x);
+
+ bits = Double.doubleToLongBits(y);
+ h_bits = getHighDWord(bits);
+ l_bits = getLowDWord(bits);
+
+ h_bits += (k << 20); // add k to y's exponent
+
+ y = buildDouble(l_bits, h_bits);
+ }
+ else
+ {
+ bits = Double.doubleToLongBits(t);
+ h_bits = (0x000003ff - k) << 20;
+ l_bits = getLowDWord(bits);
+
+ t = buildDouble(l_bits, h_bits); // t = 2^(-k)
+
+ y = x - (e + t);
+ y += 1.0;
+
+ bits = Double.doubleToLongBits(y);
+ h_bits = getHighDWord(bits);
+ l_bits = getLowDWord(bits);
+
+ h_bits += (k << 20); // add k to y's exponent
+
+ y = buildDouble(l_bits, h_bits);
+ }
+ }
+
+ return y;
+ }
+
+ /**
* Take ln(a) (the natural log). The opposite of <code>exp()</code>. If the
* argument is NaN or negative, the result is NaN; if the argument is
* positive infinity, the result is positive infinity; and if the argument
@@ -1571,6 +1819,16 @@
CBRT_G = 3.57142857142857150787e-01; // Long bits 0x3fd6db6db6db6db7L
/**
+ * Constants for computing [EMAIL PROTECTED] #expm1(double)}
+ */
+ private static final double
+ EXPM1_Q1 = -3.33333333333331316428e-02, // Long bits 0xbfa11111111110f4L
+ EXPM1_Q2 = 1.58730158725481460165e-03, // Long bits 0x3f5a01a019fe5585L
+ EXPM1_Q3 = -7.93650757867487942473e-05, // Long bits 0xbf14ce199eaadbb7L
+ EXPM1_Q4 = 4.00821782732936239552e-06, // Long bits 0x3ed0cfca86e65239L
+ EXPM1_Q5 = -2.01099218183624371326e-07; // Long bits 0xbe8afdb76e09c32dL
+
+ /**
* Helper function for reducing an angle to a multiple of pi/2 within
* [-pi/4, pi/4].
*
