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+/* cbrtl.c
+ *
+ * Cube root, long double precision
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * long double x, y, cbrtl();
+ *
+ * y = cbrtl( x );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns the cube root of the argument, which may be negative.
+ *
+ * Range reduction involves determining the power of 2 of
+ * the argument. A polynomial of degree 2 applied to the
+ * mantissa, and multiplication by the cube root of 1, 2, or 4
+ * approximates the root to within about 0.1%. Then Newton's
+ * iteration is used three times to converge to an accurate
+ * result.
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE .125,8 80000 7.0e-20 2.2e-20
+ * IEEE exp(+-707) 100000 7.0e-20 2.4e-20
+ *
+ */
+
+
+/*
+Cephes Math Library Release 2.2: January, 1991
+Copyright 1984, 1991 by Stephen L. Moshier
+Direct inquiries to 30 Frost Street, Cambridge, MA 02140
+*/
+
+
+#include <math.h>
+
+static long double CBRT2 = 1.2599210498948731647672L;
+static long double CBRT4 = 1.5874010519681994747517L;
+static long double CBRT2I = 0.79370052598409973737585L;
+static long double CBRT4I = 0.62996052494743658238361L;
+
+#ifdef ANSIPROT
+extern long double frexpl ( long double, int * );
+extern long double ldexpl ( long double, int );
+extern int isnanl ( long double );
+#else
+long double frexpl(), ldexpl();
+extern int isnanl();
+#endif
+
+#ifdef INFINITIES
+extern long double INFINITYL;
+#endif
+
+long double cbrtl(x)
+long double x;
+{
+int e, rem, sign;
+long double z;
+
+
+#ifdef NANS
+if(isnanl(x))
+ return(x);
+#endif
+#ifdef INFINITIES
+if( x == INFINITYL)
+ return(x);
+if( x == -INFINITYL)
+ return(x);
+#endif
+if( x == 0 )
+ return( x );
+if( x > 0 )
+ sign = 1;
+else
+ {
+ sign = -1;
+ x = -x;
+ }
+
+z = x;
+/* extract power of 2, leaving
+ * mantissa between 0.5 and 1
+ */
+x = frexpl( x, &e );
+
+/* Approximate cube root of number between .5 and 1,
+ * peak relative error = 1.2e-6
+ */
+x = (((( 1.3584464340920900529734e-1L * x
+ - 6.3986917220457538402318e-1L) * x
+ + 1.2875551670318751538055e0L) * x
+ - 1.4897083391357284957891e0L) * x
+ + 1.3304961236013647092521e0L) * x
+ + 3.7568280825958912391243e-1L;
+
+/* exponent divided by 3 */
+if( e >= 0 )
+ {
+ rem = e;
+ e /= 3;
+ rem -= 3*e;
+ if( rem == 1 )
+ x *= CBRT2;
+ else if( rem == 2 )
+ x *= CBRT4;
+ }
+else
+ { /* argument less than 1 */
+ e = -e;
+ rem = e;
+ e /= 3;
+ rem -= 3*e;
+ if( rem == 1 )
+ x *= CBRT2I;
+ else if( rem == 2 )
+ x *= CBRT4I;
+ e = -e;
+ }
+
+/* multiply by power of 2 */
+x = ldexpl( x, e );
+
+/* Newton iteration */
+
+x -= ( x - (z/(x*x)) )*0.3333333333333333333333L;
+x -= ( x - (z/(x*x)) )*0.3333333333333333333333L;
+
+if( sign < 0 )
+ x = -x;
+return(x);
+}