[musl] [PATCH] math: fix comments about cbrt iterations being Halley and not Newton

[musl] [PATCH] math: fix comments about cbrt iterations being Halley and not Newton

[Clément Bœsch]

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 iteration¹ 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 WolframAlpha² 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.

A similar patch has been sent and accepted³ in the FreeBSD repository
where this code originates from.

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
³ https://github.com/freebsd/freebsd-src/pull/1684

Signed-off-by: Clément Bœsch <u@pkh.me>

---
Note: I'm not subscribed to the mailing-list, so CCs are appreciated
---
 src/math/cbrt.c  | 4 ++--
 src/math/cbrtf.c | 4 ++--
 src/math/cbrtl.c | 2 +-
 3 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/math/cbrt.c b/src/math/cbrt.c
index 7599d3e3..42ac9520 100644
--- a/src/math/cbrt.c
+++ b/src/math/cbrt.c
@@ -84,7 +84,7 @@ double 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.
@@ -93,7 +93,7 @@ double cbrt(double x)
 	u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL;
 	t = u.f;
 
-	/* 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 */
diff --git a/src/math/cbrtf.c b/src/math/cbrtf.c
index 89c2c865..639c0d3e 100644
--- a/src/math/cbrtf.c
+++ b/src/math/cbrtf.c
@@ -46,7 +46,7 @@ float cbrtf(float x)
 	u.i |= hx;
 
 	/*
-	 * 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.
 	 */
@@ -55,7 +55,7 @@ float cbrtf(float x)
 	T = T*((double_t)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;
diff --git a/src/math/cbrtl.c b/src/math/cbrtl.c
index ceff9136..02d23eb5 100644
--- a/src/math/cbrtl.c
+++ b/src/math/cbrtl.c
@@ -109,7 +109,7 @@ long double 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 */
-- 
2.49.0