New private API functions _PyFloat_{Pack,Unpack}(4,8}. This is a
refactoring to get all the duplicates of this delicate code out of the
cPickle and struct modules.
diff --git a/Include/floatobject.h b/Include/floatobject.h
index 8b151cd..9a2066f 100644
--- a/Include/floatobject.h
+++ b/Include/floatobject.h
@@ -47,6 +47,48 @@
preserve precision across conversions. */
PyAPI_FUNC(void) PyFloat_AsString(char*, PyFloatObject *v);
+/* _PyFloat_{Pack,Unpack}{4,8}
+ *
+ * The struct and pickle (at least) modules need an efficient platform-
+ * independent way to store floating-point values as byte strings.
+ * The Pack routines produce a string from a C double, and the Unpack
+ * routines produce a C double from such a string. The suffix (4 or 8)
+ * specifies the number of bytes in the string.
+ *
+ * Excepting NaNs and infinities (which aren't handled correctly), the 4-
+ * byte format is identical to the IEEE-754 single precision format, and
+ * the 8-byte format to the IEEE-754 double precision format. On non-
+ * IEEE platforms with more precision, or larger dynamic range, than
+ * 754 supports, not all values can be packed; on non-IEEE platforms with
+ * less precision, or smaller dynamic range, not all values can be
+ * unpacked. What happens in such cases is partly accidental (alas).
+ */
+
+/* The pack routines write 4 or 8 bytes, starting at p. le is a bool
+ * argument, true if you want the string in little-endian format (exponent
+ * last, at p+3 or p+7), false if you want big-endian format (exponent
+ * first, at p).
+ * Return value: 0 if all is OK, -1 if error (and an exception is
+ * set, most likely OverflowError).
+ * Bug: What this does is undefined if x is a NaN or infinity.
+ * Bug: -0.0 and +0.0 produce the same string.
+ */
+PyAPI_FUNC(int) _PyFloat_Pack4(double x, unsigned char *p, int le);
+PyAPI_FUNC(int) _PyFloat_Pack8(double x, unsigned char *p, int le);
+
+/* The unpack routines read 4 or 8 bytes, starting at p. le is a bool
+ * argument, true if the string is in little-endian format (exponent
+ * last, at p+3 or p+7), false if big-endian (exponent first, at p).
+ * Return value: The unpacked double. On error, this is -1.0 and
+ * PyErr_Occurred() is true (and an exception is set, most likely
+ * OverflowError).
+ * Bug: What this does is undefined if the string represents a NaN or
+ * infinity.
+ */
+PyAPI_FUNC(double) _PyFloat_Unpack4(const unsigned char *p, int le);
+PyAPI_FUNC(double) _PyFloat_Unpack8(const unsigned char *p, int le);
+
+
#ifdef __cplusplus
}
#endif
diff --git a/Modules/cPickle.c b/Modules/cPickle.c
index 964fc63..85bcc6c 100644
--- a/Modules/cPickle.c
+++ b/Modules/cPickle.c
@@ -1124,106 +1124,10 @@
double x = PyFloat_AS_DOUBLE((PyFloatObject *)args);
if (self->bin) {
- int s, e;
- double f;
- long fhi, flo;
char str[9];
- unsigned char *p = (unsigned char *)str;
-
- *p = BINFLOAT;
- p++;
-
- if (x < 0) {
- s = 1;
- x = -x;
- }
- else
- s = 0;
-
- f = frexp(x, &e);
-
- /* Normalize f to be in the range [1.0, 2.0) */
- if (0.5 <= f && f < 1.0) {
- f *= 2.0;
- e--;
- }
- else if (f == 0.0) {
- e = 0;
- }
- else {
- PyErr_SetString(PyExc_SystemError,
- "frexp() result out of range");
+ str[0] = BINFLOAT;
+ if (_PyFloat_Pack8(x, (unsigned char *)&str[1], 0) < 0)
return -1;
- }
-
- if (e >= 1024)
- goto Overflow;
- else if (e < -1022) {
- /* Gradual underflow */
- f = ldexp(f, 1022 + e);
- e = 0;
- }
- else if (!(e == 0 && f == 0.0)) {
- e += 1023;
- f -= 1.0; /* Get rid of leading 1 */
- }
-
- /* fhi receives the high 28 bits;
- flo the low 24 bits (== 52 bits) */
- f *= 268435456.0; /* 2**28 */
- fhi = (long) floor(f); /* Truncate */
- assert(fhi < 268435456);
-
- f -= (double)fhi;
- f *= 16777216.0; /* 2**24 */
- flo = (long) floor(f + 0.5); /* Round */
- assert(flo <= 16777216);
- if (flo >> 24) {
- /* The carry propagated out of a string of 24 1 bits. */
- flo = 0;
- ++fhi;
- if (fhi >> 28) {
- /* And it also progagated out of the next
- * 28 bits.
- */
- fhi = 0;
- ++e;
- if (e >= 2047)
- goto Overflow;
- }
- }
-
- /* First byte */
- *p = (s<<7) | (e>>4);
- p++;
-
- /* Second byte */
- *p = (unsigned char) (((e&0xF)<<4) | (fhi>>24));
- p++;
-
- /* Third byte */
- *p = (unsigned char) ((fhi>>16) & 0xFF);
- p++;
-
- /* Fourth byte */
- *p = (unsigned char) ((fhi>>8) & 0xFF);
- p++;
-
- /* Fifth byte */
- *p = (unsigned char) (fhi & 0xFF);
- p++;
-
- /* Sixth byte */
- *p = (unsigned char) ((flo>>16) & 0xFF);
- p++;
-
- /* Seventh byte */
- *p = (unsigned char) ((flo>>8) & 0xFF);
- p++;
-
- /* Eighth byte */
- *p = (unsigned char) (flo & 0xFF);
-
if (self->write_func(self, str, 9) < 0)
return -1;
}
@@ -1237,11 +1141,6 @@
}
return 0;
-
- Overflow:
- PyErr_SetString(PyExc_OverflowError,
- "float too large to pack with d format");
- return -1;
}
@@ -3372,64 +3271,20 @@
static int
load_binfloat(Unpicklerobject *self)
{
- PyObject *py_float = 0;
- int s, e;
- long fhi, flo;
+ PyObject *py_float;
double x;
char *p;
if (self->read_func(self, &p, 8) < 0)
return -1;
- /* First byte */
- s = (*p>>7) & 1;
- e = (*p & 0x7F) << 4;
- p++;
+ x = _PyFloat_Unpack8((unsigned char *)p, 0);
+ if (x == -1.0 && PyErr_Occurred())
+ return -1;
- /* Second byte */
- e |= (*p>>4) & 0xF;
- fhi = (*p & 0xF) << 24;
- p++;
-
- /* Third byte */
- fhi |= (*p & 0xFF) << 16;
- p++;
-
- /* Fourth byte */
- fhi |= (*p & 0xFF) << 8;
- p++;
-
- /* Fifth byte */
- fhi |= *p & 0xFF;
- p++;
-
- /* Sixth byte */
- flo = (*p & 0xFF) << 16;
- p++;
-
- /* Seventh byte */
- flo |= (*p & 0xFF) << 8;
- p++;
-
- /* Eighth byte */
- flo |= *p & 0xFF;
-
- x = (double)fhi + (double)flo / 16777216.0; /* 2**24 */
- x /= 268435456.0; /* 2**28 */
-
- /* XXX This sadly ignores Inf/NaN */
- if (e == 0)
- e = -1022;
- else {
- x += 1.0;
- e -= 1023;
- }
- x = ldexp(x, e);
-
- if (s)
- x = -x;
-
- if (!( py_float = PyFloat_FromDouble(x))) return -1;
+ py_float = PyFloat_FromDouble(x);
+ if (py_float == NULL)
+ return -1;
PDATA_PUSH(self->stack, py_float, -1);
return 0;
diff --git a/Modules/structmodule.c b/Modules/structmodule.c
index 2210c33..e4e1eb5 100644
--- a/Modules/structmodule.c
+++ b/Modules/structmodule.c
@@ -185,301 +185,27 @@
/* Floating point helpers */
-/* These use ANSI/IEEE Standard 754-1985 (Standard for Binary Floating
- Point Arithmetic). See the following URL:
- http://www.psc.edu/general/software/packages/ieee/ieee.html */
-
-/* XXX Inf/NaN are not handled quite right (but underflow is!) */
-
-static int
-pack_float(double x, /* The number to pack */
- char *p, /* Where to pack the high order byte */
- int incr) /* 1 for big-endian; -1 for little-endian */
-{
- int s;
- int e;
- double f;
- long fbits;
-
- if (x < 0) {
- s = 1;
- x = -x;
- }
- else
- s = 0;
-
- f = frexp(x, &e);
-
- /* Normalize f to be in the range [1.0, 2.0) */
- if (0.5 <= f && f < 1.0) {
- f *= 2.0;
- e--;
- }
- else if (f == 0.0) {
- e = 0;
- }
- else {
- PyErr_SetString(PyExc_SystemError,
- "frexp() result out of range");
- return -1;
- }
-
- if (e >= 128)
- goto Overflow;
- else if (e < -126) {
- /* Gradual underflow */
- f = ldexp(f, 126 + e);
- e = 0;
- }
- else if (!(e == 0 && f == 0.0)) {
- e += 127;
- f -= 1.0; /* Get rid of leading 1 */
- }
-
- f *= 8388608.0; /* 2**23 */
- fbits = (long) floor(f + 0.5); /* Round */
- assert(fbits <= 8388608);
- if (fbits >> 23) {
- /* The carry propagated out of a string of 23 1 bits. */
- fbits = 0;
- ++e;
- if (e >= 255)
- goto Overflow;
- }
-
- /* First byte */
- *p = (s<<7) | (e>>1);
- p += incr;
-
- /* Second byte */
- *p = (char) (((e&1)<<7) | (fbits>>16));
- p += incr;
-
- /* Third byte */
- *p = (fbits>>8) & 0xFF;
- p += incr;
-
- /* Fourth byte */
- *p = fbits&0xFF;
-
- /* Done */
- return 0;
-
- Overflow:
- PyErr_SetString(PyExc_OverflowError,
- "float too large to pack with f format");
- return -1;
-}
-
-static int
-pack_double(double x, /* The number to pack */
- char *p, /* Where to pack the high order byte */
- int incr) /* 1 for big-endian; -1 for little-endian */
-{
- int s;
- int e;
- double f;
- long fhi, flo;
-
- if (x < 0) {
- s = 1;
- x = -x;
- }
- else
- s = 0;
-
- f = frexp(x, &e);
-
- /* Normalize f to be in the range [1.0, 2.0) */
- if (0.5 <= f && f < 1.0) {
- f *= 2.0;
- e--;
- }
- else if (f == 0.0) {
- e = 0;
- }
- else {
- PyErr_SetString(PyExc_SystemError,
- "frexp() result out of range");
- return -1;
- }
-
- if (e >= 1024)
- goto Overflow;
- else if (e < -1022) {
- /* Gradual underflow */
- f = ldexp(f, 1022 + e);
- e = 0;
- }
- else if (!(e == 0 && f == 0.0)) {
- e += 1023;
- f -= 1.0; /* Get rid of leading 1 */
- }
-
- /* fhi receives the high 28 bits; flo the low 24 bits (== 52 bits) */
- f *= 268435456.0; /* 2**28 */
- fhi = (long) floor(f); /* Truncate */
- assert(fhi < 268435456);
-
- f -= (double)fhi;
- f *= 16777216.0; /* 2**24 */
- flo = (long) floor(f + 0.5); /* Round */
- assert(flo <= 16777216);
- if (flo >> 24) {
- /* The carry propagated out of a string of 24 1 bits. */
- flo = 0;
- ++fhi;
- if (fhi >> 28) {
- /* And it also progagated out of the next 28 bits. */
- fhi = 0;
- ++e;
- if (e >= 2047)
- goto Overflow;
- }
- }
-
- /* First byte */
- *p = (s<<7) | (e>>4);
- p += incr;
-
- /* Second byte */
- *p = (char) (((e&0xF)<<4) | (fhi>>24));
- p += incr;
-
- /* Third byte */
- *p = (fhi>>16) & 0xFF;
- p += incr;
-
- /* Fourth byte */
- *p = (fhi>>8) & 0xFF;
- p += incr;
-
- /* Fifth byte */
- *p = fhi & 0xFF;
- p += incr;
-
- /* Sixth byte */
- *p = (flo>>16) & 0xFF;
- p += incr;
-
- /* Seventh byte */
- *p = (flo>>8) & 0xFF;
- p += incr;
-
- /* Eighth byte */
- *p = flo & 0xFF;
- p += incr;
-
- /* Done */
- return 0;
-
- Overflow:
- PyErr_SetString(PyExc_OverflowError,
- "float too large to pack with d format");
- return -1;
-}
-
static PyObject *
-unpack_float(const char *p, /* Where the high order byte is */
- int incr) /* 1 for big-endian; -1 for little-endian */
+unpack_float(const char *p, /* start of 4-byte string */
+ int le) /* true for little-endian, false for big-endian */
{
- int s;
- int e;
- long f;
double x;
- /* First byte */
- s = (*p>>7) & 1;
- e = (*p & 0x7F) << 1;
- p += incr;
-
- /* Second byte */
- e |= (*p>>7) & 1;
- f = (*p & 0x7F) << 16;
- p += incr;
-
- /* Third byte */
- f |= (*p & 0xFF) << 8;
- p += incr;
-
- /* Fourth byte */
- f |= *p & 0xFF;
-
- x = (double)f / 8388608.0;
-
- /* XXX This sadly ignores Inf/NaN issues */
- if (e == 0)
- e = -126;
- else {
- x += 1.0;
- e -= 127;
- }
- x = ldexp(x, e);
-
- if (s)
- x = -x;
-
+ x = _PyFloat_Unpack4((unsigned char *)p, le);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
return PyFloat_FromDouble(x);
}
static PyObject *
-unpack_double(const char *p, /* Where the high order byte is */
- int incr) /* 1 for big-endian; -1 for little-endian */
+unpack_double(const char *p, /* start of 8-byte string */
+ int le) /* true for little-endian, false for big-endian */
{
- int s;
- int e;
- long fhi, flo;
double x;
- /* First byte */
- s = (*p>>7) & 1;
- e = (*p & 0x7F) << 4;
- p += incr;
-
- /* Second byte */
- e |= (*p>>4) & 0xF;
- fhi = (*p & 0xF) << 24;
- p += incr;
-
- /* Third byte */
- fhi |= (*p & 0xFF) << 16;
- p += incr;
-
- /* Fourth byte */
- fhi |= (*p & 0xFF) << 8;
- p += incr;
-
- /* Fifth byte */
- fhi |= *p & 0xFF;
- p += incr;
-
- /* Sixth byte */
- flo = (*p & 0xFF) << 16;
- p += incr;
-
- /* Seventh byte */
- flo |= (*p & 0xFF) << 8;
- p += incr;
-
- /* Eighth byte */
- flo |= *p & 0xFF;
- p += incr;
-
- x = (double)fhi + (double)flo / 16777216.0; /* 2**24 */
- x /= 268435456.0; /* 2**28 */
-
- /* XXX This sadly ignores Inf/NaN */
- if (e == 0)
- e = -1022;
- else {
- x += 1.0;
- e -= 1023;
- }
- x = ldexp(x, e);
-
- if (s)
- x = -x;
-
+ x = _PyFloat_Unpack8((unsigned char *)p, le);
+ if (x == -1.0 && PyErr_Occurred())
+ return NULL;
return PyFloat_FromDouble(x);
}
@@ -887,13 +613,13 @@
static PyObject *
bu_float(const char *p, const formatdef *f)
{
- return unpack_float(p, 1);
+ return unpack_float(p, 0);
}
static PyObject *
bu_double(const char *p, const formatdef *f)
{
- return unpack_double(p, 1);
+ return unpack_double(p, 0);
}
static int
@@ -967,7 +693,7 @@
"required argument is not a float");
return -1;
}
- return pack_float(x, p, 1);
+ return _PyFloat_Pack4(x, (unsigned char *)p, 0);
}
static int
@@ -979,7 +705,7 @@
"required argument is not a float");
return -1;
}
- return pack_double(x, p, 1);
+ return _PyFloat_Pack8(x, (unsigned char *)p, 0);
}
static formatdef bigendian_table[] = {
@@ -1053,13 +779,13 @@
static PyObject *
lu_float(const char *p, const formatdef *f)
{
- return unpack_float(p+3, -1);
+ return unpack_float(p, 1);
}
static PyObject *
lu_double(const char *p, const formatdef *f)
{
- return unpack_double(p+7, -1);
+ return unpack_double(p, 1);
}
static int
@@ -1133,7 +859,7 @@
"required argument is not a float");
return -1;
}
- return pack_float(x, p+3, -1);
+ return _PyFloat_Pack4(x, (unsigned char *)p, 1);
}
static int
@@ -1145,7 +871,7 @@
"required argument is not a float");
return -1;
}
- return pack_double(x, p+7, -1);
+ return _PyFloat_Pack8(x, (unsigned char *)p, 1);
}
static formatdef lilendian_table[] = {
diff --git a/Objects/floatobject.c b/Objects/floatobject.c
index 6e65756..3cbc98a 100644
--- a/Objects/floatobject.c
+++ b/Objects/floatobject.c
@@ -904,3 +904,316 @@
}
}
}
+
+/*----------------------------------------------------------------------------
+ * _PyFloat_{Pack,Unpack}{4,8}. See floatobject.h.
+ *
+ * TODO: On platforms that use the standard IEEE-754 single and double
+ * formats natively, these routines could simply copy the bytes.
+ */
+int
+_PyFloat_Pack4(double x, unsigned char *p, int le)
+{
+ unsigned char sign;
+ int e;
+ double f;
+ unsigned int fbits;
+ int incr = 1;
+
+ if (le) {
+ p += 3;
+ incr = -1;
+ }
+
+ if (x < 0) {
+ sign = 1;
+ x = -x;
+ }
+ else
+ sign = 0;
+
+ f = frexp(x, &e);
+
+ /* Normalize f to be in the range [1.0, 2.0) */
+ if (0.5 <= f && f < 1.0) {
+ f *= 2.0;
+ e--;
+ }
+ else if (f == 0.0)
+ e = 0;
+ else {
+ PyErr_SetString(PyExc_SystemError,
+ "frexp() result out of range");
+ return -1;
+ }
+
+ if (e >= 128)
+ goto Overflow;
+ else if (e < -126) {
+ /* Gradual underflow */
+ f = ldexp(f, 126 + e);
+ e = 0;
+ }
+ else if (!(e == 0 && f == 0.0)) {
+ e += 127;
+ f -= 1.0; /* Get rid of leading 1 */
+ }
+
+ f *= 8388608.0; /* 2**23 */
+ fbits = (long) floor(f + 0.5); /* Round */
+ assert(fbits <= 8388608);
+ if (fbits >> 23) {
+ /* The carry propagated out of a string of 23 1 bits. */
+ fbits = 0;
+ ++e;
+ if (e >= 255)
+ goto Overflow;
+ }
+
+ /* First byte */
+ *p = (sign << 7) | (e >> 1);
+ p += incr;
+
+ /* Second byte */
+ *p = (char) (((e & 1) << 7) | (fbits >> 16));
+ p += incr;
+
+ /* Third byte */
+ *p = (fbits >> 8) & 0xFF;
+ p += incr;
+
+ /* Fourth byte */
+ *p = fbits & 0xFF;
+
+ /* Done */
+ return 0;
+
+ Overflow:
+ PyErr_SetString(PyExc_OverflowError,
+ "float too large to pack with f format");
+ return -1;
+}
+
+int
+_PyFloat_Pack8(double x, unsigned char *p, int le)
+{
+ unsigned char sign;
+ int e;
+ double f;
+ unsigned int fhi, flo;
+ int incr = 1;
+
+ if (le) {
+ p += 7;
+ incr = -1;
+ }
+
+ if (x < 0) {
+ sign = 1;
+ x = -x;
+ }
+ else
+ sign = 0;
+
+ f = frexp(x, &e);
+
+ /* Normalize f to be in the range [1.0, 2.0) */
+ if (0.5 <= f && f < 1.0) {
+ f *= 2.0;
+ e--;
+ }
+ else if (f == 0.0)
+ e = 0;
+ else {
+ PyErr_SetString(PyExc_SystemError,
+ "frexp() result out of range");
+ return -1;
+ }
+
+ if (e >= 1024)
+ goto Overflow;
+ else if (e < -1022) {
+ /* Gradual underflow */
+ f = ldexp(f, 1022 + e);
+ e = 0;
+ }
+ else if (!(e == 0 && f == 0.0)) {
+ e += 1023;
+ f -= 1.0; /* Get rid of leading 1 */
+ }
+
+ /* fhi receives the high 28 bits; flo the low 24 bits (== 52 bits) */
+ f *= 268435456.0; /* 2**28 */
+ fhi = (unsigned int)f; /* Truncate */
+ assert(fhi < 268435456);
+
+ f -= (double)fhi;
+ f *= 16777216.0; /* 2**24 */
+ flo = (unsigned int)(f + 0.5); /* Round */
+ assert(flo <= 16777216);
+ if (flo >> 24) {
+ /* The carry propagated out of a string of 24 1 bits. */
+ flo = 0;
+ ++fhi;
+ if (fhi >> 28) {
+ /* And it also progagated out of the next 28 bits. */
+ fhi = 0;
+ ++e;
+ if (e >= 2047)
+ goto Overflow;
+ }
+ }
+
+ /* First byte */
+ *p = (sign << 7) | (e >> 4);
+ p += incr;
+
+ /* Second byte */
+ *p = (unsigned char) (((e & 0xF) << 4) | (fhi >> 24));
+ p += incr;
+
+ /* Third byte */
+ *p = (fhi >> 16) & 0xFF;
+ p += incr;
+
+ /* Fourth byte */
+ *p = (fhi >> 8) & 0xFF;
+ p += incr;
+
+ /* Fifth byte */
+ *p = fhi & 0xFF;
+ p += incr;
+
+ /* Sixth byte */
+ *p = (flo >> 16) & 0xFF;
+ p += incr;
+
+ /* Seventh byte */
+ *p = (flo >> 8) & 0xFF;
+ p += incr;
+
+ /* Eighth byte */
+ *p = flo & 0xFF;
+ p += incr;
+
+ /* Done */
+ return 0;
+
+ Overflow:
+ PyErr_SetString(PyExc_OverflowError,
+ "float too large to pack with d format");
+ return -1;
+}
+
+double
+_PyFloat_Unpack4(const unsigned char *p, int le)
+{
+ unsigned char sign;
+ int e;
+ unsigned int f;
+ double x;
+ int incr = 1;
+
+ if (le) {
+ p += 3;
+ incr = -1;
+ }
+
+ /* First byte */
+ sign = (*p >> 7) & 1;
+ e = (*p & 0x7F) << 1;
+ p += incr;
+
+ /* Second byte */
+ e |= (*p >> 7) & 1;
+ f = (*p & 0x7F) << 16;
+ p += incr;
+
+ /* Third byte */
+ f |= *p << 8;
+ p += incr;
+
+ /* Fourth byte */
+ f |= *p;
+
+ x = (double)f / 8388608.0;
+
+ /* XXX This sadly ignores Inf/NaN issues */
+ if (e == 0)
+ e = -126;
+ else {
+ x += 1.0;
+ e -= 127;
+ }
+ x = ldexp(x, e);
+
+ if (sign)
+ x = -x;
+
+ return x;
+}
+
+double
+_PyFloat_Unpack8(const unsigned char *p, int le)
+{
+ unsigned char sign;
+ int e;
+ unsigned int fhi, flo;
+ double x;
+ int incr = 1;
+
+ if (le) {
+ p += 7;
+ incr = -1;
+ }
+
+ /* First byte */
+ sign = (*p >> 7) & 1;
+ e = (*p & 0x7F) << 4;
+ p += incr;
+
+ /* Second byte */
+ e |= (*p >> 4) & 0xF;
+ fhi = (*p & 0xF) << 24;
+ p += incr;
+
+ /* Third byte */
+ fhi |= *p << 16;
+ p += incr;
+
+ /* Fourth byte */
+ fhi |= *p << 8;
+ p += incr;
+
+ /* Fifth byte */
+ fhi |= *p;
+ p += incr;
+
+ /* Sixth byte */
+ flo = *p << 16;
+ p += incr;
+
+ /* Seventh byte */
+ flo |= *p << 8;
+ p += incr;
+
+ /* Eighth byte */
+ flo |= *p;
+
+ x = (double)fhi + (double)flo / 16777216.0; /* 2**24 */
+ x /= 268435456.0; /* 2**28 */
+
+ /* XXX This sadly ignores Inf/NaN */
+ if (e == 0)
+ e = -1022;
+ else {
+ x += 1.0;
+ e -= 1023;
+ }
+ x = ldexp(x, e);
+
+ if (sign)
+ x = -x;
+
+ return x;
+}