math: fix sinh overflows in non-nearest rounding

The final rounding operation should be done with the correct sign
otherwise huge results may incorrectly get rounded to or away from
infinity in upward or downward rounding modes.

This affected sinh and sinhf which set the sign on the result after
a potentially overflowing mul. There may be other non-nearest rounding
issues, but this was a known long standing issue with large ulp error
(depending on how ulp is defined near infinity).

The fix should have no effect on sinh and sinhf performance but may
have a tiny effect on cosh and coshf.

src/internal/libm.h

@@ -236,13 +236,13 @@ hidden int    __rem_pio2(double,double*);
 hidden double __sin(double,double,int);
 hidden double __cos(double,double);
 hidden double __tan(double,double,int);
-hidden double __expo2(double);
+hidden double __expo2(double,double);
 
 hidden int    __rem_pio2f(float,double*);
 hidden float  __sindf(double);
 hidden float  __cosdf(double);
 hidden float  __tandf(double,int);
-hidden float  __expo2f(float);
+hidden float  __expo2f(float,float);
 
 hidden int __rem_pio2l(long double, long double *);
 hidden long double __sinl(long double, long double, int);

src/math/__expo2.c

@@ -5,12 +5,13 @@ static const int k = 2043;
 static const double kln2 = 0x1.62066151add8bp+10;
 
 /* exp(x)/2 for x >= log(DBL_MAX), slightly better than 0.5*exp(x/2)*exp(x/2) */
-double __expo2(double x)
+double __expo2(double x, double sign)
 {
 	double scale;
 
 	/* note that k is odd and scale*scale overflows */
 	INSERT_WORDS(scale, (uint32_t)(0x3ff + k/2) << 20, 0);
 	/* exp(x - k ln2) * 2**(k-1) */
-	return exp(x - kln2) * scale * scale;
+	/* in directed rounding correct sign before rounding or overflow is important */
+	return exp(x - kln2) * (sign * scale) * scale;
 }

src/math/__expo2f.c

@@ -5,12 +5,13 @@ static const int k = 235;
 static const float kln2 = 0x1.45c778p+7f;
 
 /* expf(x)/2 for x >= log(FLT_MAX), slightly better than 0.5f*expf(x/2)*expf(x/2) */
-float __expo2f(float x)
+float __expo2f(float x, float sign)
 {
 	float scale;
 
 	/* note that k is odd and scale*scale overflows */
 	SET_FLOAT_WORD(scale, (uint32_t)(0x7f + k/2) << 23);
 	/* exp(x - k ln2) * 2**(k-1) */
-	return expf(x - kln2) * scale * scale;
+	/* in directed rounding correct sign before rounding or overflow is important */
+	return expf(x - kln2) * (sign * scale) * scale;
 }

src/math/cosh.c

@@ -35,6 +35,6 @@ double cosh(double x)
 
 	/* |x| > log(DBL_MAX) or nan */
 	/* note: the result is stored to handle overflow */
-	t = __expo2(x);
+	t = __expo2(x, 1.0);
 	return t;
 }

src/math/coshf.c

@@ -28,6 +28,6 @@ float coshf(float x)
 	}
 
 	/* |x| > log(FLT_MAX) or nan */
-	t = __expo2f(x);
+	t = __expo2f(x, 1.0f);
 	return t;
 }

src/math/sinh.c

@@ -34,6 +34,6 @@ double sinh(double x)
 
 	/* |x| > log(DBL_MAX) or nan */
 	/* note: the result is stored to handle overflow */
-	t = 2*h*__expo2(absx);
+	t = __expo2(absx, 2*h);
 	return t;
 }

src/math/sinhf.c

@@ -26,6 +26,6 @@ float sinhf(float x)
 	}
 
 	/* |x| > logf(FLT_MAX) or nan */
-	t = 2*h*__expo2f(absx);
+	t = __expo2f(absx, 2*h);
 	return t;
 }