changeset 42751:38d28e784f44

8139688: Port fdlibm exp to Java Reviewed-by: bpb, nadezhin
author darcy
date Fri, 16 Dec 2016 21:43:29 -0800
parents 562fee8337a3
children 844691c73832
files jdk/make/mapfiles/libjava/mapfile-vers jdk/src/java.base/share/classes/java/lang/FdLibm.java jdk/src/java.base/share/classes/java/lang/StrictMath.java jdk/src/java.base/share/native/libjava/StrictMath.c jdk/test/java/lang/StrictMath/ExpTests.java jdk/test/java/lang/StrictMath/FdlibmTranslit.java
diffstat 6 files changed, 441 insertions(+), 15 deletions(-) [+]
line wrap: on
line diff
--- a/jdk/make/mapfiles/libjava/mapfile-vers	Fri Dec 16 11:58:17 2016 -0800
+++ b/jdk/make/mapfiles/libjava/mapfile-vers	Fri Dec 16 21:43:29 2016 -0800
@@ -150,7 +150,6 @@
 		Java_java_lang_StrictMath_atan;
 		Java_java_lang_StrictMath_atan2;
 		Java_java_lang_StrictMath_cos;
-		Java_java_lang_StrictMath_exp;
 		Java_java_lang_StrictMath_log;
 		Java_java_lang_StrictMath_log10;
 		Java_java_lang_StrictMath_sin;
--- a/jdk/src/java.base/share/classes/java/lang/FdLibm.java	Fri Dec 16 11:58:17 2016 -0800
+++ b/jdk/src/java.base/share/classes/java/lang/FdLibm.java	Fri Dec 16 21:43:29 2016 -0800
@@ -1,5 +1,5 @@
 /*
- * Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved.
+ * Copyright (c) 1998, 2016, Oracle and/or its affiliates. All rights reserved.
  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
  *
  * This code is free software; you can redistribute it and/or modify it
@@ -79,7 +79,8 @@
      */
     private static double __LO(double x, int low) {
         long transX = Double.doubleToRawLongBits(x);
-        return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
+        return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) |
+                                       (low    & 0x0000_0000_FFFF_FFFFL));
     }
 
     /**
@@ -96,7 +97,8 @@
      */
     private static double __HI(double x, int high) {
         long transX = Double.doubleToRawLongBits(x);
-        return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
+        return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) |
+                                       ( ((long)high)) << 32 );
     }
 
     /**
@@ -580,4 +582,152 @@
             return s * z;
         }
     }
+
+    /**
+     * Returns the exponential of x.
+     *
+     * Method
+     *   1. Argument reduction:
+     *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
+     *      Given x, find r and integer k such that
+     *
+     *               x = k*ln2 + r,  |r| <= 0.5*ln2.
+     *
+     *      Here r will be represented as r = hi-lo for better
+     *      accuracy.
+     *
+     *   2. Approximation of exp(r) by a special rational function on
+     *      the interval [0,0.34658]:
+     *      Write
+     *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
+     *      We use a special Reme algorithm on [0,0.34658] to generate
+     *      a polynomial of degree 5 to approximate R. The maximum error
+     *      of this polynomial approximation is bounded by 2**-59. In
+     *      other words,
+     *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
+     *      (where z=r*r, and the values of P1 to P5 are listed below)
+     *      and
+     *          |                  5          |     -59
+     *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
+     *          |                             |
+     *      The computation of exp(r) thus becomes
+     *                             2*r
+     *              exp(r) = 1 + -------
+     *                            R - r
+     *                                 r*R1(r)
+     *                     = 1 + r + ----------- (for better accuracy)
+     *                                2 - R1(r)
+     *      where
+     *                               2       4             10
+     *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
+     *
+     *   3. Scale back to obtain exp(x):
+     *      From step 1, we have
+     *         exp(x) = 2^k * exp(r)
+     *
+     * Special cases:
+     *      exp(INF) is INF, exp(NaN) is NaN;
+     *      exp(-INF) is 0, and
+     *      for finite argument, only exp(0)=1 is exact.
+     *
+     * Accuracy:
+     *      according to an error analysis, the error is always less than
+     *      1 ulp (unit in the last place).
+     *
+     * Misc. info.
+     *      For IEEE double
+     *          if x >  7.09782712893383973096e+02 then exp(x) overflow
+     *          if x < -7.45133219101941108420e+02 then exp(x) underflow
+     *
+     * Constants:
+     * The hexadecimal values are the intended ones for the following
+     * constants. The decimal values may be used, provided that the
+     * compiler will convert from decimal to binary accurately enough
+     * to produce the hexadecimal values shown.
+     */
+    static class Exp {
+        private static final double one     = 1.0;
+        private static final double[] half = {0.5, -0.5,};
+        private static final double huge    = 1.0e+300;
+        private static final double twom1000=     0x1.0p-1000;             //  9.33263618503218878990e-302 = 2^-1000
+        private static final double o_threshold=  0x1.62e42fefa39efp9;     //  7.09782712893383973096e+02
+        private static final double u_threshold= -0x1.74910d52d3051p9;     // -7.45133219101941108420e+02;
+        private static final double[] ln2HI   ={  0x1.62e42feep-1,         //  6.93147180369123816490e-01
+                                                 -0x1.62e42feep-1};        // -6.93147180369123816490e-01
+        private static final double[] ln2LO   ={  0x1.a39ef35793c76p-33,   //  1.90821492927058770002e-10
+                                                 -0x1.a39ef35793c76p-33};  // -1.90821492927058770002e-10
+        private static final double invln2 =      0x1.71547652b82fep0;     //  1.44269504088896338700e+00
+
+        private static final double P1   =  0x1.555555555553ep-3;  //  1.66666666666666019037e-01
+        private static final double P2   = -0x1.6c16c16bebd93p-9;  // -2.77777777770155933842e-03
+        private static final double P3   =  0x1.1566aaf25de2cp-14; //  6.61375632143793436117e-05
+        private static final double P4   = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06
+        private static final double P5   =  0x1.6376972bea4d0p-25; //  4.13813679705723846039e-08
+
+        // should be able to forgo strictfp due to controlled over/underflow
+        public static strictfp double compute(double x) {
+            double y;
+            double hi = 0.0;
+            double lo = 0.0;
+            double c;
+            double t;
+            int k = 0;
+            int xsb;
+            /*unsigned*/ int hx;
+
+            hx  = __HI(x);  /* high word of x */
+            xsb = (hx >> 31) & 1;               /* sign bit of x */
+            hx &= 0x7fffffff;               /* high word of |x| */
+
+            /* filter out non-finite argument */
+            if (hx >= 0x40862E42) {                  /* if |x| >= 709.78... */
+                if (hx >= 0x7ff00000) {
+                    if (((hx & 0xfffff) | __LO(x)) != 0)
+                        return x + x;                /* NaN */
+                    else
+                        return (xsb == 0) ? x : 0.0;    /* exp(+-inf) = {inf, 0} */
+                }
+                if (x > o_threshold)
+                    return huge * huge; /* overflow */
+                if (x < u_threshold) // unsigned compare needed here?
+                    return twom1000 * twom1000; /* underflow */
+            }
+
+            /* argument reduction */
+            if (hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
+                if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
+                    hi = x - ln2HI[xsb];
+                    lo=ln2LO[xsb];
+                    k = 1 - xsb - xsb;
+                } else {
+                    k  = (int)(invln2 * x + half[xsb]);
+                    t  = k;
+                    hi = x - t*ln2HI[0];    /* t*ln2HI is exact here */
+                    lo = t*ln2LO[0];
+                }
+                x  = hi - lo;
+            } else if (hx < 0x3e300000)  {     /* when |x|<2**-28 */
+                if (huge + x > one)
+                    return one + x; /* trigger inexact */
+            } else {
+                k = 0;
+            }
+
+            /* x is now in primary range */
+            t  = x * x;
+            c  = x - t*(P1 + t*(P2 + t*(P3 + t*(P4 + t*P5))));
+            if (k == 0)
+                return one - ((x*c)/(c - 2.0) - x);
+            else
+                y = one - ((lo - (x*c)/(2.0 - c)) - hi);
+
+            if(k >= -1021) {
+                y = __HI(y, __HI(y) + (k << 20)); /* add k to y's exponent */
+                return y;
+            } else {
+                y = __HI(y, __HI(y) + ((k + 1000) << 20)); /* add k to y's exponent */
+                return y * twom1000;
+            }
+        }
+    }
 }
--- a/jdk/src/java.base/share/classes/java/lang/StrictMath.java	Fri Dec 16 11:58:17 2016 -0800
+++ b/jdk/src/java.base/share/classes/java/lang/StrictMath.java	Fri Dec 16 21:43:29 2016 -0800
@@ -227,7 +227,9 @@
      * @return  the value <i>e</i><sup>{@code a}</sup>,
      *          where <i>e</i> is the base of the natural logarithms.
      */
-    public static native double exp(double a);
+    public static double exp(double a) {
+        return FdLibm.Exp.compute(a);
+    }
 
     /**
      * Returns the natural logarithm (base <i>e</i>) of a {@code double}
--- a/jdk/src/java.base/share/native/libjava/StrictMath.c	Fri Dec 16 11:58:17 2016 -0800
+++ b/jdk/src/java.base/share/native/libjava/StrictMath.c	Fri Dec 16 21:43:29 2016 -0800
@@ -1,5 +1,5 @@
 /*
- * Copyright (c) 1994, 2015, Oracle and/or its affiliates. All rights reserved.
+ * Copyright (c) 1994, 2016, Oracle and/or its affiliates. All rights reserved.
  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
  *
  * This code is free software; you can redistribute it and/or modify it
@@ -65,12 +65,6 @@
 }
 
 JNIEXPORT jdouble JNICALL
-Java_java_lang_StrictMath_exp(JNIEnv *env, jclass unused, jdouble d)
-{
-    return (jdouble) jexp((double)d);
-}
-
-JNIEXPORT jdouble JNICALL
 Java_java_lang_StrictMath_log(JNIEnv *env, jclass unused, jdouble d)
 {
     return (jdouble) jlog((double)d);
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/jdk/test/java/lang/StrictMath/ExpTests.java	Fri Dec 16 21:43:29 2016 -0800
@@ -0,0 +1,147 @@
+/*
+ * Copyright (c) 2015, 2016, Oracle and/or its affiliates. All rights reserved.
+ * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
+ *
+ * This code is free software; you can redistribute it and/or modify it
+ * under the terms of the GNU General Public License version 2 only, as
+ * published by the Free Software Foundation.
+ *
+ * This code is distributed in the hope that it will be useful, but WITHOUT
+ * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+ * FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
+ * version 2 for more details (a copy is included in the LICENSE file that
+ * accompanied this code).
+ *
+ * You should have received a copy of the GNU General Public License version
+ * 2 along with this work; if not, write to the Free Software Foundation,
+ * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
+ *
+ * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
+ * or visit www.oracle.com if you need additional information or have any
+ * questions.
+ */
+
+/*
+ * @test
+ * @bug 8139688
+ * @key randomness
+ * @library /lib/testlibrary/
+ * @build jdk.testlibrary.RandomFactory
+ * @build Tests
+ * @build FdlibmTranslit
+ * @build ExpTests
+ * @run main ExpTests
+ * @summary Tests specifically for StrictMath.exp
+ */
+
+import jdk.testlibrary.RandomFactory;
+
+/**
+ * The role of this test is to verify that the FDLIBM exp algorithm is
+ * being used by running golden file style tests on values that may
+ * vary from one conforming exponential implementation to another.
+ */
+
+public class ExpTests {
+    private ExpTests(){}
+
+    public static void main(String [] argv) {
+        int failures = 0;
+
+        failures += testExp();
+        failures += testAgainstTranslit();
+
+        if (failures > 0) {
+            System.err.println("Testing the exponential incurred "
+                               + failures + " failures.");
+            throw new RuntimeException();
+        }
+    }
+
+    // From the fdlibm source, the overflow threshold in hex is:
+    // 0x4086_2E42_FEFA_39EF.
+    static final double EXP_OVERFLOW_THRESH  = Double.longBitsToDouble(0x4086_2E42_FEFA_39EFL);
+
+    // From the fdlibm source, the underflow threshold in hex is:
+    // 0xc087_4910_D52D_3051L.
+    static final double EXP_UNDERFLOW_THRESH = Double.longBitsToDouble(0xc087_4910_D52D_3051L);
+
+    static int testExp() {
+        int failures = 0;
+
+        double [][] testCases = {
+            // Some of these could be moved to common Math/StrictMath exp testing.
+            {Double.NaN,                      Double.NaN},
+            {Double.MAX_VALUE,                Double.POSITIVE_INFINITY},
+            {Double.POSITIVE_INFINITY,        Double.POSITIVE_INFINITY},
+            {Double.NEGATIVE_INFINITY,        +0.0},
+            {EXP_OVERFLOW_THRESH,                 0x1.ffff_ffff_fff2ap1023},
+            {Math.nextUp(EXP_OVERFLOW_THRESH),    Double.POSITIVE_INFINITY},
+            {Math.nextDown(EXP_UNDERFLOW_THRESH), +0.0},
+            {EXP_UNDERFLOW_THRESH,                +Double.MIN_VALUE},
+        };
+
+        for(double[] testCase: testCases)
+            failures+=testExpCase(testCase[0], testCase[1]);
+
+        return failures;
+    }
+
+    static int testExpCase(double input, double expected) {
+        int failures = 0;
+
+        failures+=Tests.test("StrictMath.exp(double)", input,
+                             StrictMath.exp(input), expected);
+        return failures;
+    }
+
+    // Initialize shared random number generator
+    private static java.util.Random random = RandomFactory.getRandom();
+
+    /**
+     * Test StrictMath.exp against transliteration port of exp.
+     */
+    private static int testAgainstTranslit() {
+        int failures = 0;
+
+        double[] decisionPoints = {
+            // Near overflow threshold
+            EXP_OVERFLOW_THRESH - 512*Math.ulp(EXP_OVERFLOW_THRESH),
+
+            // Near underflow threshold
+            EXP_UNDERFLOW_THRESH - 512*Math.ulp(EXP_UNDERFLOW_THRESH),
+
+            // Straddle algorithm conditional checks
+            Double.longBitsToDouble(0x4086_2E42_0000_0000L - 512L),
+            Double.longBitsToDouble(0x3fd6_2e42_0000_0000L - 512L),
+            Double.longBitsToDouble(0x3FF0_A2B2_0000_0000L - 512L),
+            Double.longBitsToDouble(0x3e30_0000_0000_0000L - 512L),
+
+            // Other notable points
+            Double.MIN_NORMAL - Math.ulp(Double.MIN_NORMAL)*512,
+            -Double.MIN_VALUE*512,
+        };
+
+        for (double decisionPoint : decisionPoints) {
+            double ulp = Math.ulp(decisionPoint);
+            failures += testRange(decisionPoint - 1024*ulp, ulp, 1_024);
+        }
+
+        // Try out some random values
+        for (int i = 0; i < 100; i++) {
+            double x = Tests.createRandomDouble(random);
+            failures += testRange(x, Math.ulp(x), 100);
+        }
+
+        return failures;
+    }
+
+    private static int testRange(double start, double increment, int count) {
+        int failures = 0;
+        double x = start;
+        for (int i = 0; i < count; i++, x += increment) {
+            failures += testExpCase(x, FdlibmTranslit.Exp.compute(x));
+        }
+        return failures;
+    }
+}
--- a/jdk/test/java/lang/StrictMath/FdlibmTranslit.java	Fri Dec 16 11:58:17 2016 -0800
+++ b/jdk/test/java/lang/StrictMath/FdlibmTranslit.java	Fri Dec 16 21:43:29 2016 -0800
@@ -1,5 +1,5 @@
 /*
- * Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved.
+ * Copyright (c) 1998, 2016, Oracle and/or its affiliates. All rights reserved.
  * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
  *
  * This code is free software; you can redistribute it and/or modify it
@@ -48,7 +48,8 @@
      */
     private static double __LO(double x, int low) {
         long transX = Double.doubleToRawLongBits(x);
-        return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
+        return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) |
+                                       (low    & 0x0000_0000_FFFF_FFFFL));
     }
 
     /**
@@ -65,7 +66,8 @@
      */
     private static double __HI(double x, int high) {
         long transX = Double.doubleToRawLongBits(x);
-        return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
+        return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) |
+                                       ( ((long)high)) << 32 );
     }
 
     public static double hypot(double x, double y) {
@@ -250,4 +252,136 @@
                 return w;
         }
     }
+
+    /**
+     * Returns the exponential of x.
+     *
+     * Method
+     *   1. Argument reduction:
+     *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
+     *      Given x, find r and integer k such that
+     *
+     *               x = k*ln2 + r,  |r| <= 0.5*ln2.
+     *
+     *      Here r will be represented as r = hi-lo for better
+     *      accuracy.
+     *
+     *   2. Approximation of exp(r) by a special rational function on
+     *      the interval [0,0.34658]:
+     *      Write
+     *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
+     *      We use a special Reme algorithm on [0,0.34658] to generate
+     *      a polynomial of degree 5 to approximate R. The maximum error
+     *      of this polynomial approximation is bounded by 2**-59. In
+     *      other words,
+     *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
+     *      (where z=r*r, and the values of P1 to P5 are listed below)
+     *      and
+     *          |                  5          |     -59
+     *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
+     *          |                             |
+     *      The computation of exp(r) thus becomes
+     *                             2*r
+     *              exp(r) = 1 + -------
+     *                            R - r
+     *                                 r*R1(r)
+     *                     = 1 + r + ----------- (for better accuracy)
+     *                                2 - R1(r)
+     *      where
+     *                               2       4             10
+     *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
+     *
+     *   3. Scale back to obtain exp(x):
+     *      From step 1, we have
+     *         exp(x) = 2^k * exp(r)
+     *
+     * Special cases:
+     *      exp(INF) is INF, exp(NaN) is NaN;
+     *      exp(-INF) is 0, and
+     *      for finite argument, only exp(0)=1 is exact.
+     *
+     * Accuracy:
+     *      according to an error analysis, the error is always less than
+     *      1 ulp (unit in the last place).
+     *
+     * Misc. info.
+     *      For IEEE double
+     *          if x >  7.09782712893383973096e+02 then exp(x) overflow
+     *          if x < -7.45133219101941108420e+02 then exp(x) underflow
+     *
+     * Constants:
+     * The hexadecimal values are the intended ones for the following
+     * constants. The decimal values may be used, provided that the
+     * compiler will convert from decimal to binary accurately enough
+     * to produce the hexadecimal values shown.
+     */
+    static class Exp {
+        private static final double one     = 1.0;
+        private static final double[] halF = {0.5,-0.5,};
+        private static final double huge    = 1.0e+300;
+        private static final double twom1000= 9.33263618503218878990e-302;      /* 2**-1000=0x01700000,0*/
+        private static final double o_threshold=  7.09782712893383973096e+02;   /* 0x40862E42, 0xFEFA39EF */
+        private static final double u_threshold= -7.45133219101941108420e+02;   /* 0xc0874910, 0xD52D3051 */
+        private static final double[] ln2HI   ={ 6.93147180369123816490e-01,    /* 0x3fe62e42, 0xfee00000 */
+                                                 -6.93147180369123816490e-01};  /* 0xbfe62e42, 0xfee00000 */
+        private static final double[] ln2LO   ={ 1.90821492927058770002e-10,    /* 0x3dea39ef, 0x35793c76 */
+                                                 -1.90821492927058770002e-10,}; /* 0xbdea39ef, 0x35793c76 */
+        private static final double invln2 =  1.44269504088896338700e+00;       /* 0x3ff71547, 0x652b82fe */
+        private static final double P1   =  1.66666666666666019037e-01;         /* 0x3FC55555, 0x5555553E */
+        private static final double P2   = -2.77777777770155933842e-03;         /* 0xBF66C16C, 0x16BEBD93 */
+        private static final double P3   =  6.61375632143793436117e-05;         /* 0x3F11566A, 0xAF25DE2C */
+        private static final double P4   = -1.65339022054652515390e-06;         /* 0xBEBBBD41, 0xC5D26BF1 */
+        private static final double P5   =  4.13813679705723846039e-08;         /* 0x3E663769, 0x72BEA4D0 */
+
+        public static strictfp double compute(double x) {
+            double y,hi=0,lo=0,c,t;
+            int k=0,xsb;
+            /*unsigned*/ int hx;
+
+            hx  = __HI(x);  /* high word of x */
+            xsb = (hx>>31)&1;               /* sign bit of x */
+            hx &= 0x7fffffff;               /* high word of |x| */
+
+            /* filter out non-finite argument */
+            if(hx >= 0x40862E42) {                  /* if |x|>=709.78... */
+                if(hx>=0x7ff00000) {
+                    if(((hx&0xfffff)|__LO(x))!=0)
+                        return x+x;                /* NaN */
+                    else return (xsb==0)? x:0.0;    /* exp(+-inf)={inf,0} */
+                }
+                if(x > o_threshold) return huge*huge; /* overflow */
+                if(x < u_threshold) return twom1000*twom1000; /* underflow */
+            }
+
+            /* argument reduction */
+            if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */
+                if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
+                    hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
+                } else {
+                    k  = (int)(invln2*x+halF[xsb]);
+                    t  = k;
+                    hi = x - t*ln2HI[0];    /* t*ln2HI is exact here */
+                    lo = t*ln2LO[0];
+                }
+                x  = hi - lo;
+            }
+            else if(hx < 0x3e300000)  {     /* when |x|<2**-28 */
+                if(huge+x>one) return one+x;/* trigger inexact */
+            }
+            else k = 0;
+
+            /* x is now in primary range */
+            t  = x*x;
+            c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
+            if(k==0)        return one-((x*c)/(c-2.0)-x);
+            else            y = one-((lo-(x*c)/(2.0-c))-hi);
+            if(k >= -1021) {
+                y = __HI(y, __HI(y) + (k<<20)); /* add k to y's exponent */
+                return y;
+            } else {
+                y = __HI(y, __HI(y) + ((k+1000)<<20));/* add k to y's exponent */
+                return y*twom1000;
+            }
+        }
+    }
 }