msun: fix cbrt iterations from Newton to Halley method

Since we're inverting a cube, we have:
    f(Tₙ)=Tₙ³-x    (1)

Its first and second derivatives are:
    f'(Tₙ)=3Tₙ²    (2)
    f"(Tₙ)=6Tₙ     (3)

Halley iteration[1] uses:
    Tₙ₊₁=Tₙ-2f(Tₙ)f'(Tₙ)/(2f'(Tₙ)²-f(Tₙ)f"(Tₙ))    (4)

Replacing the terms of (4) using (1), (2) and (3):

Tₙ₊₁ = Tₙ-2f(Tₙ)f'(Tₙ)/(2f'(Tₙ)²-f(Tₙ)f"(Tₙ))
     = Tₙ-2(Tₙ³-x)3Tₙ²/(2(3Tₙ²)²-(Tₙ³-x)6Tₙ)
     = <snip, see WolframAlpha[2] alternate forms>
     = Tₙ(2x+Tₙ³)/(x+2Tₙ³)

This formula corresponds to the exact expression used in the code.

Newton formula is Tₙ-f(Tₙ)/f'(Tₙ) which would have simplified to
(2Tₙ³+x)/(3Tₙ²) instead.

[1] https://en.wikipedia.org/wiki/Halley's_method
[2] https://www.wolframalpha.com/input?i=T-2%28T%5E3-x%293T%5E2%2F%282%283T%5E2%29%5E2-%28T%5E3-x%296T%29

Note: UTF8 in commit message due to the heavy math being hard to
recreate w/o it. -- imp

Signed-off-by: Clément Bœsch <u@pkh.me>
Reviewed by: imp
Pull Request: https://github.com/freebsd/freebsd-src/pull/1684

lib/msun/src/s_cbrt.c

@@ -90,7 +90,7 @@ cbrt(double x)
      * the result is larger in magnitude than cbrt(x) but not much more than
      * 2 23-bit ulps larger).  With rounding towards zero, the error bound
      * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
-     * in the rounded t, the infinite-precision error in the Newton
+     * in the rounded t, the infinite-precision error in the Halley
      * approximation barely affects third digit in the final error
      * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
      * before the final error is larger than 0.667 ulps.
@@ -99,7 +99,7 @@ cbrt(double x)
 	u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
 	t=u.value;
 
-    /* one step Newton iteration to 53 bits with error < 0.667 ulps */
+    /* one step Halley iteration to 53 bits with error < 0.667 ulps */
 	s=t*t;				/* t*t is exact */
 	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
 	w=t+t;				/* t+t is exact */

lib/msun/src/s_cbrtf.c

@@ -50,7 +50,7 @@ cbrtf(float x)
 	    SET_FLOAT_WORD(t,sign|(hx/3+B1));
 
     /*
-     * First step Newton iteration (solving t*t-x/t == 0) to 16 bits.  In
+     * First step Halley iteration (solving t*t-x/t == 0) to 16 bits.  In
      * double precision so that its terms can be arranged for efficiency
      * without causing overflow or underflow.
      */
@@ -59,7 +59,7 @@ cbrtf(float x)
 	T=T*((double)x+x+r)/(x+r+r);
 
     /*
-     * Second step Newton iteration to 47 bits.  In double precision for
+     * Second step Halley iteration to 47 bits.  In double precision for
      * efficiency and accuracy.
      */
 	r=T*T*T;

lib/msun/src/s_cbrtl.c

@@ -126,7 +126,7 @@ cbrtl(long double x)
 #endif
 
 	/*
-     	 * Final step Newton iteration to 64 or 113 bits with
+	 * Final step Halley iteration to 64 or 113 bits with
 	 * error < 0.667 ulps
 	 */
 	s=t*t;				/* t*t is exact */