Fix error bounds for fast pass.

Author: lntue

libc/src/math/generic/acos.cpp

@@ -40,7 +40,7 @@ LLVM_LIBC_FUNCTION(double, acos, (double x)) {
 #else
       // Force the evaluation and prevent constant propagation so that it
       // is rounded correctly for FE_UPWARD rounding mode.
-      return fputil::multiply_add(xbits.abs().get_val(), 0.75, PI_OVER_TWO.hi);
+      return (xbits.abs().get_val() + 0x1.0p-160) + PI_OVER_TWO.hi;
 #endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
     }
 
@@ -52,10 +52,9 @@ LLVM_LIBC_FUNCTION(double, acos, (double x)) {
 #else
     unsigned idx;
     DoubleDouble x_sq = fputil::exact_mult(x, x);
-    double err = x * 0x1.0p-51;
+    double err = xbits.abs().get_val() * 0x1.0p-51;
     // Polynomial approximation:
     //   p ~ asin(x)/x
-
     DoubleDouble p = asin_eval(x_sq, idx, err);
     // asin(x) ~ x * p
     DoubleDouble r0 = fputil::exact_mult(x, p.hi);

libc/src/math/generic/asin.cpp

@@ -74,7 +74,7 @@ LLVM_LIBC_FUNCTION(double, asin, (double x)) {
 #else
     unsigned idx;
     DoubleDouble x_sq = fputil::exact_mult(x, x);
-    double err = x * 0x1.0p-51;
+    double err = xbits.abs().get_val() * 0x1.0p-51;
     // Polynomial approximation:
     //   p ~ asin(x)/x
 

libc/src/math/generic/asin_utils.h

@@ -174,7 +174,8 @@ LIBC_INLINE DoubleDouble asin_eval(const DoubleDouble &u, unsigned &idx,
   double y_hi = multiply_add(k, -0x1.0p-5, u.hi); // Exact
   DoubleDouble y = fputil::exact_add(y_hi, u.lo);
   double y2 = y.hi * y.hi;
-  err *= y2 + 0x1.0p-30;
+  // Add double-double errors in addition to the relative errors from y2.
+  err = fputil::multiply_add(err, y2, 0x1.0p-102);
   DoubleDouble c0 = fputil::quick_mult(
       y, DoubleDouble{ASIN_COEFFS[idx][3], ASIN_COEFFS[idx][2]});
   double c1 = multiply_add(y.hi, ASIN_COEFFS[idx][5], ASIN_COEFFS[idx][4]);