math: fix asin, atan, log1p, tanh to raise underflow on subnormal
for these functions f(x)=x for small inputs, because f(0)=0 and
f'(0)=1, but for subnormal values they should raise the underflow
flag (required by annex F), if they are approximated by a polynomial
around 0 then spurious underflow should be avoided (not required by
annex F)
all these functions should raise inexact flag for small x if x!=0,
but it's not required by the standard and it does not seem a worthy
goal, so support for it is removed in some cases.
raising underflow:
- x*x may not raise underflow for subnormal x if FLT_EVAL_METHOD!=0
- x*x may raise spurious underflow for normal x if FLT_EVAL_METHOD==0
- in case of double subnormal x, store x as float
- in case of float subnormal x, store x*x as float
diff --git a/src/math/asin.c b/src/math/asin.c
index 3e8f99e..c926b18 100644
--- a/src/math/asin.c
+++ b/src/math/asin.c
@@ -82,11 +82,9 @@
}
/* |x| < 0.5 */
if (ix < 0x3fe00000) {
- if (ix < 0x3e500000) {
- /* |x|<0x1p-26, return x with inexact if x!=0*/
- FORCE_EVAL(x + 0x1p120f);
+ /* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */
+ if (ix < 0x3e500000 && ix >= 0x00100000)
return x;
- }
return x + x*R(x*x);
}
/* 1 > |x| >= 0.5 */
diff --git a/src/math/asinf.c b/src/math/asinf.c
index 51fe6c6..bcd304a 100644
--- a/src/math/asinf.c
+++ b/src/math/asinf.c
@@ -46,10 +46,9 @@
return 0/(x-x); /* asin(|x|>1) is NaN */
}
if (ix < 0x3f000000) { /* |x| < 0.5 */
- if (ix < 0x39800000) { /* |x| < 2**-12 */
- FORCE_EVAL(x + 0x1p120f);
- return x; /* return x with inexact if x!=0 */
- }
+ /* if 0x1p-126 <= |x| < 0x1p-12, avoid raising underflow */
+ if (ix < 0x39800000 && ix >= 0x00800000)
+ return x;
return x + x*R(x*x);
}
/* 1 > |x| >= 0.5 */
diff --git a/src/math/atan.c b/src/math/atan.c
index 5a1d33e..63b0ab2 100644
--- a/src/math/atan.c
+++ b/src/math/atan.c
@@ -77,8 +77,9 @@
}
if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
if (ix < 0x3e400000) { /* |x| < 2^-27 */
- /* raise inexact if x!=0 */
- FORCE_EVAL(x + 0x1p120f);
+ if (ix < 0x00100000)
+ /* raise underflow for subnormal x */
+ FORCE_EVAL((float)x);
return x;
}
id = -1;
diff --git a/src/math/atanf.c b/src/math/atanf.c
index ac8bfd0..178341b 100644
--- a/src/math/atanf.c
+++ b/src/math/atanf.c
@@ -55,8 +55,9 @@
}
if (ix < 0x3ee00000) { /* |x| < 0.4375 */
if (ix < 0x39800000) { /* |x| < 2**-12 */
- /* raise inexact if x!=0 */
- FORCE_EVAL(x + 0x1p120f);
+ if (ix < 0x00800000)
+ /* raise underflow for subnormal x */
+ FORCE_EVAL(x*x);
return x;
}
id = -1;
diff --git a/src/math/log1p.c b/src/math/log1p.c
index 6c67249..9bed63c 100644
--- a/src/math/log1p.c
+++ b/src/math/log1p.c
@@ -104,9 +104,12 @@
return (x-x)/(x-x); /* log1p(x<-1)=NaN */
}
if (ax < 0x3e200000) { /* |x| < 2**-29 */
- /* raise inexact */
- if (two54 + x > 0.0 && ax < 0x3c900000) /* |x| < 2**-54 */
+ /* if 0x1p-1022 <= |x| < 0x1p-54, avoid raising underflow */
+ if (ax < 0x3c900000 && ax >= 0x00100000)
return x;
+#if FLT_EVAL_METHOD != 0
+ FORCE_EVAL((float)x);
+#endif
return x - x*x*0.5;
}
if (hx > 0 || hx <= (int32_t)0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
diff --git a/src/math/log1pf.c b/src/math/log1pf.c
index 39832d2..c38e0bc 100644
--- a/src/math/log1pf.c
+++ b/src/math/log1pf.c
@@ -43,9 +43,12 @@
return (x-x)/(x-x); /* log1p(x<-1)=NaN */
}
if (ax < 0x38000000) { /* |x| < 2**-15 */
- /* raise inexact */
- if (two25 + x > 0.0f && ax < 0x33800000) /* |x| < 2**-24 */
+ /* if 0x1p-126 <= |x| < 0x1p-24, avoid raising underflow */
+ if (ax < 0x33800000 && ax >= 0x00800000)
return x;
+#if FLT_EVAL_METHOD != 0
+ FORCE_EVAL(x*x);
+#endif
return x - x*x*0.5f;
}
if (hx > 0 || hx <= (int32_t)0xbe95f619) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
diff --git a/src/math/sinh.c b/src/math/sinh.c
index 47e36bf..00022c4 100644
--- a/src/math/sinh.c
+++ b/src/math/sinh.c
@@ -23,8 +23,8 @@
t = expm1(absx);
if (w < 0x3ff00000) {
if (w < 0x3ff00000 - (26<<20))
- /* note: inexact is raised by expm1 */
- /* note: this branch avoids underflow */
+ /* note: inexact and underflow are raised by expm1 */
+ /* note: this branch avoids spurious underflow */
return x;
return h*(2*t - t*t/(t+1));
}
diff --git a/src/math/tanh.c b/src/math/tanh.c
index 0e766c5..20d6dbc 100644
--- a/src/math/tanh.c
+++ b/src/math/tanh.c
@@ -9,7 +9,7 @@
union {double f; uint64_t i;} u = {.f = x};
uint32_t w;
int sign;
- double t;
+ double_t t;
/* x = |x| */
sign = u.i >> 63;
@@ -22,8 +22,7 @@
if (w > 0x40340000) {
/* |x| > 20 or nan */
/* note: this branch avoids raising overflow */
- /* raise inexact if x!=+-inf and handle nan */
- t = 1 + 0/(x + 0x1p-120f);
+ t = 1 - 0/x;
} else {
t = expm1(2*x);
t = 1 - 2/(t+2);
@@ -32,10 +31,15 @@
/* |x| > log(5/3)/2 ~= 0.2554 */
t = expm1(2*x);
t = t/(t+2);
- } else {
- /* |x| is small, up to 2ulp error in [0.1,0.2554] */
+ } else if (w >= 0x00100000) {
+ /* |x| >= 0x1p-1022, up to 2ulp error in [0.1,0.2554] */
t = expm1(-2*x);
t = -t/(t+2);
+ } else {
+ /* |x| is subnormal */
+ /* note: the branch above would not raise underflow in [0x1p-1023,0x1p-1022) */
+ FORCE_EVAL((float)x);
+ t = x;
}
return sign ? -t : t;
}
diff --git a/src/math/tanhf.c b/src/math/tanhf.c
index 8099ec3..10636fb 100644
--- a/src/math/tanhf.c
+++ b/src/math/tanhf.c
@@ -17,7 +17,7 @@
/* |x| > log(3)/2 ~= 0.5493 or nan */
if (w > 0x41200000) {
/* |x| > 10 */
- t = 1 + 0/(x + 0x1p-120f);
+ t = 1 + 0/x;
} else {
t = expm1f(2*x);
t = 1 - 2/(t+2);
@@ -26,10 +26,14 @@
/* |x| > log(5/3)/2 ~= 0.2554 */
t = expm1f(2*x);
t = t/(t+2);
- } else {
- /* |x| is small */
+ } else if (w >= 0x00800000) {
+ /* |x| >= 0x1p-126 */
t = expm1f(-2*x);
t = -t/(t+2);
+ } else {
+ /* |x| is subnormal */
+ FORCE_EVAL(x*x);
+ t = x;
}
return sign ? -t : t;
}