fix several bugs and formatting in the fast math ulp tables (#544)

* fix several bugs and formatting in the fast math ulp tables

* add back OpFDiv to environment spec description
unify formatting

Author: bashbaug

OpenCL_C.txt

@@ -11301,7 +11301,7 @@ there was sufficient range, meets ULP error tolerance.
 | _x_ + _y_             | Correctly rounded
 | _x_ - _y_             | Correctly rounded
 | _x_ * _y_             | Correctly rounded
-| *1.0 / _x_*           | {leq} 2.5 ulp
+| 1.0 / _x_             | {leq} 2.5 ulp
 | _x_ / _y_             | {leq} 2.5 ulp
 | |
 | *acos*                | {leq} 4 ulp
@@ -11487,7 +11487,7 @@ is the infinitely precise result.
 | _x_ + _y_         | Correctly rounded
 | _x_ - _y_         | Correctly rounded
 | _x_ * _y_         | Correctly rounded
-| *1.0 / _x_*       | {leq} 3 ulp
+| 1.0 / _x_         | {leq} 3 ulp
 | _x_ / _y_         | {leq} 3 ulp
 | |
 | *acos*            | {leq} 4 ulp
@@ -11625,136 +11625,188 @@ requires>> support for OpenCL C 2.0 or newer.
 
 [[table-float-ulp-relaxed]]
 .ULP values for single precision built-in math functions with unsafe math optimizations in the full and embedded profiles
-[cols=",",]
+[cols="3,7",]
 |====
-| *Function* | *Min Accuracy - ULP values*
-| *1.0 / _x_*
+| *Function*
+| *Minimum Accuracy*
+
+| 1.0 / _x_
     | {leq} 2.5 ulp for _x_ in the domain of 2^-126^ to 2^126^ for the full
       profile, and {leq} 3 ulp for the embedded profile.
+
 | _x_ / _y_
     | {leq} 2.5 ulp for _x_ in the domain of 2^-62^ to 2^62^ and _y_ in the
       domain of 2^-62^ to 2^62^ for the full profile, and {leq} 3 ulp for
       the embedded profile.
+
 | *acos*(_x_)
     | {leq} 4096 ulp
+
+| *acosh*(_x_)
+    | Derived implementations may implement as *log*(_x_ + *sqrt*(_x_ * _x_ - 1)).
+
 | *acospi*(_x_)
-    | Implemented as *acos*(_x_) * `M_PI_F`.
+    | Derived implementations may implement as *acos*(_x_) * `M_PI_F`.
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *asin*(_x_)
     | {leq} 4096 ulp
+
+| *asinh*(_x_)
+    | Derived implementations may implement as *log*(_x_ + *sqrt*(_x_ * _x_ + 1)).
+
 | *asinpi*(_x_)
-    | Implemented as *asin*(_x_) * `M_PI_F`.
+    | Derived implementations may implement as *asin*(_x_) * `M_PI_F`.
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *atan*(_x_)
     | {leq} 4096 ulp
-| *atan2*(_y_, _x_)
-    | Implemented as *atan*(_y_ / _x_) for _x_ > 0, *atan*(_y_ / _x_) +
-      `M_1_PI_F` for _x_ < 0 and _y_ > 0 and *atan*(_y_ / _x_) -
-      `M_1_PI_F` for _x_ < 0 and _y_ < 0.
+
+//| *atanh*(_x_)
+//  | Defined for _x_ in the domain (-1, 1).
+//    For _x_ in [-2^-10^, 2^-10^], derived implementations may implement as _x_.
+//    For _x_ outside of [-2^-10^, 2^-10^], derived implementations may implement as
+//    0.5f * *log*((1.0f + _x_) / (1.0f - _x_)).
+//    For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *atanpi*(_x_)
-    | Implemented as *atan*(_x_) * `M_1_PI_F`.
+    | Derived implementations may implement as *atan*(_x_) * `M_1_PI_F`.
       For non-derived implementations, the error is {leq} 8192 ulp.
+
+| *atan2*(_y_, _x_)
+    | Derived implementations may implement as *atan*(_y_ / _x_) for _x_ > 0,
+      *atan*(_y_ / _x_) + `M_PI_F` for _x_ < 0 and _y_ > 0, and
+      *atan*(_y_ / _x_) - `M_PI_F` for _x_ < 0 and _y_ < 0.
+
 | *atan2pi*(_y_, _x_)
-    | Implemented as *atan2*(_y_, _x_) * `M_PI_F`.
+    | Derived implementations may implement as *atan2*(_y_, _x_) * `M_1_PI_F`.
       For non-derived implementations, the error is {leq} 8192 ulp.
-| *acosh*(_x_)
-    | Implemented as *log*(_x_ + *sqrt*(_x_ * _x_ - 1)).
-| *asinh*(_x_)
-    | Implemented as *log*(_x_ + *sqrt*(_x_ * _x_ + 1)).
+
 | *cbrt*(_x_)
-    | Implemented as *rootn*(_x_, 3).
+    | Derived implementations may implement as *rootn*(_x_, 3).
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *cos*(_x_)
     | For _x_ in the domain [-{pi}, {pi}], the maximum absolute error
       is {leq} 2^-11^ and larger otherwise.
+
 | *cosh*(_x_)
-    | Defined for _x_ in the domain [-88,88] and implemented as 0.5f *
-      (*exp*(_x_) + *exp*(-_x_)).
+    | Defined for _x_ in the domain [-88, 88].
+      Derived implementations may implement as 0.5f * (*exp*(_x_) + *exp*(-_x_)).
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *cospi*(_x_)
     | For _x_ in the domain [-1, 1], the maximum absolute error is {leq}
       2^-11^ and larger otherwise.
+
 | *exp*(_x_)
     | {leq} 3 + *floor*(*fabs*(2 * _x_)) ulp for the full profile, and {leq}
       4 ulp for the embedded profile.
+
 | *exp2*(_x_)
     | {leq} 3 + *floor*(*fabs*(2 * _x_)) ulp for the full profile, and {leq}
       4 ulp for the embedded profile.
+
 | *exp10*(_x_)
-    | Derived implementations implement this as *exp2*(_x_ * *log2*(10)).
+    | Derived implementations may implement as *exp2*(_x_ * *log2*(10)).
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *expm1*(_x_)
-    | Derived implementations implement this as *exp*(_x_) - 1.
+    | Derived implementations may implement as *exp*(_x_) - 1.
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *log*(_x_)
     | For _x_ in the domain [0.5, 2] the maximum absolute error is {leq}
-      2^-21^; otherwise the maximum error is {leq}3 ulp for the full profile
-      and {leq} 4 ulp for the embedded profile
+      2^-21^; otherwise the maximum error is {leq} 3 ulp for the full profile
+      and {leq} 4 ulp for the embedded profile.
+
 | *log2*(_x_)
     | For _x_ in the domain [0.5, 2] the maximum absolute error is {leq}
-      2^-21^; otherwise the maximum error is {leq}3 ulp for the full profile
-      and {leq} 4 ulp for the embedded profile
+      2^-21^; otherwise the maximum error is {leq} 3 ulp for the full profile
+      and {leq} 4 ulp for the embedded profile.
+
 | *log10*(_x_)
     | For _x_ in the domain [0.5, 2] the maximum absolute error is {leq}
-      2^-21^; otherwise the maximum error is {leq}3 ulp for the full profile
-      and {leq} 4 ulp for the embedded profile
+      2^-21^; otherwise the maximum error is {leq} 3 ulp for the full profile
+      and {leq} 4 ulp for the embedded profile.
+
 | *log1p*(_x_)
-    | Derived implementations implement this as *log*(_x_ + 1).
+    | Derived implementations may implement as *log*(_x_ + 1).
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *pow*(_x_, _y_)
     | Undefined for _x_ = 0 and _y_ = 0.
-      Undefined for _x_ < 0 and non-integer y.
-      Undefined for _x_ < 0 and _y_ outside the domain [-2^24, 2^24].
-      For _x_ > 0 or _x_ < 0 and even _y_, derived implementations implement
-      this as *exp2*(_y_ * *log2*(*fabs*(_x_))).
-      For _x_ < 0 and odd _y_, derived implementations implement this as
+      Undefined for _x_ < 0 and non-integer _y_.
+      Undefined for _x_ < 0 and _y_ outside the domain [-2^24^, 2^24^].
+      For _x_ > 0 or _x_ < 0 and even _y_, derived implementations may implement as
+      *exp2*(_y_ * *log2*(*fabs*(_x_))).
+      For _x_ < 0 and odd _y_, derived implementations may implement as
       -*exp2*(_y_ * *log2*(*fabs*(_x_)).
-      For _x_ == 0 and nonzero _y_, derived implementations return zero.
+      For _x_ == 0 and non-zero _y_, for derived implementations may return zero.
       For non-derived implementations, the error is {leq} 8192 ulp.
       footnote:[{fn-pow-performance}]
+
 | *pown*(_x_, _y_)
-    | Defined only for integer values of y.
+    | Defined only for integer values of _y_.
       Undefined for _x_ = 0 and _y_ = 0.
-      For _x_ >= 0 or _x_ < 0 and even _y_, derived implementations
-      implement this as *exp2*(_y_ * *log2*(*fabs*(_x_))).
-      For _x_ < 0 and odd _y_, derived implementations implement this as
-      -*exp2*(_y_ * *log2*(*fabs*(_x_)).
+      For _x_ >= 0 or _x_ < 0 and even _y_, derived implementations may implement as
+      *exp2*(_y_ * *log2*(*fabs*(_x_))).
+      For _x_ < 0 and odd _y_, derived implementations may implement as
+      -*exp2*(_y_ * *log2*(*fabs*(_x_))).
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *powr*(_x_, _y_)
     | Defined only for _x_ >= 0.
       Undefined for _x_ = 0 and _y_ = 0.
-      Derived implementations implement this as *exp2*(_y_ * *log2*(_x_)).
+      Derived implementations may implement as *exp2*(_y_ * *log2*(_x_)).
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *rootn*(_x_, _y_)
-    | Defined for _x_ > 0 when _y_ is nonzero, derived implementations
-      implement this case as *exp2*(log2(_x_) / _y_).
-      Defined for _x_ < 0 when _y_ is odd, derived implementations implement
-      this case as -*exp2*(*log2*(-_x_) / _y_).
-      Defined for _x_ = +/-0 when _y_ > 0, derived implementations will
+    | Defined for _x_ > 0 when _y_ is non-zero, derived implementations
+      may implement this case as *exp2*(*log2*(_x_) / _y_).
+      Defined for _x_ < 0 when _y_ is odd, derived implementations
+      may implement this case as -*exp2*(*log2*(-_x_) / _y_).
+      Defined for _x_ = +/-0 when _y_ > 0, derived implementations may
       return +0 in this case.
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *sin*(_x_)
     | For _x_ in the domain [-{pi}, {pi}], the maximum absolute error is
       {leq} 2^-11^ and larger otherwise.
+
 | *sincos*(_x_)
-    | ulp values as defined for *sin*(_x_) and *cos*(_x_)
+    | ulp values as defined for *sin*(_x_) and *cos*(_x_).
+
 | *sinh*(_x_)
     | Defined for _x_ in the domain [-88,88].
-      For _x_ in [-2^-10,2^-10], derived implementations implement as _x_.
-      For _x_ outside of [-2^10,2^10], derived implement as *0.5f *
-      (*exp*(_x_) - *exp*(-_x_)).
+      For _x_ in [-2^-10^, 2^-10^], derived implementations
+      may implement as _x_.
+      For _x_ outside of [-2^-10^, 2^-10^], derived implementations
+      may implement as 0.5f * (*exp*(_x_) - *exp*(-_x_)).
       For non-derived implementations, the error is {leq} 8192 ulp.
+
 | *sinpi*(_x_)
     | For _x_ in the domain [-1, 1], the maximum absolute error is {leq}
       2^-11^ and larger otherwise.
+
 | *tan*(_x_)
-    | Derived implementations implement this as *sin*(_x_) * (`1.0f` /
-      *cos*(_x_)).
+    | Derived implementations may implement as
+      *sin*(_x_) * (1.0f / *cos*(_x_)).
       For non-derived implementations, the error is {leq} 8192 ulp.
+
+//| *tanh*(_x_)
+//  | Defined for _x_ in the domain [-{inf}, {inf}].
+//    For _x_ in [-2^-10^, 2^-10^], derived implementations
+//    may implement as _x_.
+//    For _x_ outside of [-2^-10^, 2^-10^], derived implementations
+//    may implement as (*exp*(_x_) - *exp*(-_x_)) / (*exp*(_x_) + *exp*(-_x_)).
+//    For non-derived implementations, the error is {leq} 8192 ULP.
+
 | *tanpi*(_x_)
-    | Derived implementations implement this as *tan*(_x_ * `M_PI_F`).
+    | Derived implementations may implement as *tan*(_x_ * `M_PI_F`).
       For non-derived implementations, the error is {leq} 8192 ulp for _x_
       in the domain [-1, 1].
+
 | _x_ * _y_ + _z_
     | Implemented either as a correctly rounded *fma* or as a multiply and
       an add both of which are correctly rounded.