Improve precision of FSQRT - #42

Test output after the change:

Running test of square root(x).
Testing if sqrt(X * X) == X for 20 Integers X.
Test for sqrt monotonicity.
sqrt has passed a test for Monotonicity.
Testing whether sqrt is rounded or chopped.
Square root is neither chopped nor correctly rounded.
Observed errors run from 0.E0 to 5.000000000000000000E-1  ulps.

lib/~ik/float-ieee-binary/f-sqrt.4th

@@ -1,4 +1,5 @@
-64 CONSTANT FSQRT-ITERATIONS-DEFAULT
+32 CONSTANT FSQRT-ITERATIONS-DEFAULT
+
 
 : FSQRT-APPROX (F r1 -- r2 )
    \G r2 is the initial approximation of square root of r1.
@@ -8,31 +9,48 @@
    'FPX FPE!
 ;
 
+
 : FSQRT-NEWTON-STEP (F r1 r2 -- r3 )
    \G Perform step in Newton approximation for a square root.
    \G Calculate next approximation r3 for the square root of
    \G r1 given the previous approximation r2.
-   ?FPX0= IF  FNIP EXIT  THEN
+   ?FPX0=  IF  FNIP EXIT  THEN
    FSWAP FOVER      \ F: r2 r1 r2
    F/ F+
-   FPX2/
+   FTWO F/
 ;
 
-: FSQRT-NEWTON (S +n -- ) (F r1 r2 -- r3 )
+
+: FSQRT-NEWTON-STEP2 (F r1 r2 -- r3 )
+   \G Perform step in Newton approximation for a square root.
+   \G Calculate next approximation r3 for the square root of
+   \G r1 given the previous approximation r2.
+   ?FPX0=  IF  FNIP EXIT  THEN
+   FSWAP FOVER FDUP F*    \ F: r2 r1 r2**2
+   FSWAP F-               \ F: r2 r2**2-r1
+   FOVER FTWO F* F/       \ F: r2 (r2**2-r1)/(2*r2)
+   F-
+;
+
+
+: FSQRT-NEWTON (S +n xt -- ) (F r1 r2 -- r3 )
    \G Calculate square root of r1 given the initial approximation r2 and number of iterations +n.
+   \G xt calculates next approximation with stack effect (F r1 r2 -- r3 ).
    1 ?FPSTACK-OVERFLOW
+   >R
    BEGIN
       DUP 0>
       ?FPX0= INVERT \ stop iterations if approximation eq. zero
       AND
    WHILE
       FOVER FSWAP
       \DEBUG S" FSQRT-NEWTON-A1: " CR TYPE CR F.DUMP CR
-      FSQRT-NEWTON-STEP
+      R@ EXECUTE
       \DEBUG S" FSQRT-NEWTON-A2: " CR TYPE CR F.DUMP CR
       1-
    REPEAT
    DROP
+   R> DROP
    FNIP
    \DEBUG S" FSQRT-NEWTON-B: " CR TYPE CR F.DUMP CR
 ;
@@ -45,5 +63,6 @@
    ?FPX0=    IF  EXIT  THEN
    ?FPX0<    IF  FPX-NAN! EXIT  THEN
    ?FPX-INF  IF  EXIT  THEN
-   FSQRT-ITERATIONS-DEFAULT FDUP FSQRT-APPROX FSQRT-NEWTON
+   FDUP FSQRT-APPROX
+   FSQRT-ITERATIONS-DEFAULT ['] FSQRT-NEWTON-STEP2 FSQRT-NEWTON
 ;