[libc][math] Improve the error analysis and accuracy for pow function. (#102098)

1 files changed, 45 insertions, 23 deletions

diff --git a/libc/src/math/generic/pow.cpp b/libc/src/math/generic/pow.cpp
index d2c5f7535082..20f914430261 100644
--- a/libc/src/math/generic/pow.cpp
+++ b/libc/src/math/generic/pow.cpp
@@ -358,42 +358,64 @@ LLVM_LIBC_FUNCTION(double, pow, (double x, double y)) {
   // Then m_x = (1 + dx) / r, and
   //   log2(m_x) = log2( (1 + dx) / r )
   //             = log2(1 + dx) - log2(r).
-  // Perform exact range reduction
+
+  // In order for the overall computations x^y = 2^(y * log2(x)) to have the
+  // relative errors < 2^-52 (1ULP), we will need to evaluate the exponent part
+  // y * log2(x) with absolute errors < 2^-52 (or better, 2^-53).  Since the
+  // whole exponent range for double precision is bounded by
+  // |y * log2(x)| < 1076 ~ 2^10, we need to evaluate log2(x) with absolute
+  // errors < 2^-53 * 2^-10 = 2^-63.
+
+  // With that requirement, we use the following degree-6 polynomial
+  // approximation:
+  //   P(dx) ~ log2(1 + dx) / dx
+  // Generated by Sollya with:
+  // > P = fpminimax(log2(1 + x)/x, 6, [|D...|], [-2^-8, 2^-7]); P;
+  // > dirtyinfnorm(log2(1 + x) - x*P, [-2^-8, 2^-7]);
+  //   0x1.d03cc...p-66
+  constexpr double COEFFS[] = {0x1.71547652b82fep0,  -0x1.71547652b82e7p-1,
+                               0x1.ec709dc3b1fd5p-2, -0x1.7154766124215p-2,
+                               0x1.2776bd90259d8p-2, -0x1.ec586c6f3d311p-3,
+                               0x1.9c4775eccf524p-3};
+  // Error: ulp(dx^2) <= (2^-7)^2 * 2^-52 = 2^-66
+  // Extra errors from various computations and rounding directions, the overall
+  // errors we can be bounded by 2^-65.
+
   double dx;
+  DoubleDouble dx_c0;
+
+  // Perform exact range reduction and exact product dx * c0.
 #ifdef LIBC_TARGET_CPU_HAS_FMA
   dx = fputil::multiply_add(RD[idx_x], m_x.get_val(), -1.0); // Exact
+  dx_c0 = fputil::exact_mult(COEFFS[0], dx);
 #else
   double c = FPBits(m_x.uintval() & 0x3fff'e000'0000'0000).get_val();
   dx = fputil::multiply_add(RD[idx_x], m_x.get_val() - c, CD[idx_x]); // Exact
+  dx_c0 = fputil::exact_mult<true>(COEFFS[0], dx);
 #endif // LIBC_TARGET_CPU_HAS_FMA
 
-  // Degree-5 polynomial approximation:
-  //   dx * P(dx) ~ log2(1 + dx)
-  // Generated by Sollya with:
-  // > P = fpminimax(log2(1 + x)/x, 5, [|D...|], [-2^-8, 2^-7]);
-  // > dirtyinfnorm(log2(1 + x)/x - P, [-2^-8, 2^-7]);
-  //   0x1.653...p-52
-  constexpr double COEFFS[] = {0x1.71547652b82fep0,  -0x1.71547652b7a07p-1,
-                               0x1.ec709dc458db1p-2, -0x1.715479c2266c9p-2,
-                               0x1.2776ae1ddf8fp-2,  -0x1.e7b2178870157p-3};
-
-  double dx2 = dx * dx; // Exact
-  double c0 = fputil::multiply_add(dx, COEFFS[1], COEFFS[0]);
-  double c1 = fputil::multiply_add(dx, COEFFS[3], COEFFS[2]);
-  double c2 = fputil::multiply_add(dx, COEFFS[5], COEFFS[4]);
+  double dx2 = dx * dx;
+  double c0 = fputil::multiply_add(dx, COEFFS[2], COEFFS[1]);
+  double c1 = fputil::multiply_add(dx, COEFFS[4], COEFFS[3]);
+  double c2 = fputil::multiply_add(dx, COEFFS[6], COEFFS[5]);
 
   double p = fputil::polyeval(dx2, c0, c1, c2);
 
   // s = e_x - log2(r) + dx * P(dx)
   // Absolute error bound:
-  //   |log2(x) - log2_x.hi - log2_x.lo| < 2^-58.
-  // Relative error bound:
-  //   |(log2_x.hi + log2_x.lo)/log2(x) - 1| < 2^-51.
-  double log2_x_hi = e_x + LOG2_R_DD[idx_x].hi; // Exact
-  // Error
-  double log2_x_lo = fputil::multiply_add(dx, p, LOG2_R_DD[idx_x].lo);
-
-  DoubleDouble log2_x = fputil::exact_add(log2_x_hi, log2_x_lo);
+  //   |log2(x) - log2_x.hi - log2_x.lo| < 2^-65.
+
+  // Notice that e_x - log2(r).hi is exact, so we perform an exact sum of
+  // e_x - log2(r).hi and the high part of the product dx * c0:
+  //   log2_x_hi.hi + log2_x_hi.lo = e_x - log2(r).hi + (dx * c0).hi
+  DoubleDouble log2_x_hi =
+      fputil::exact_add(e_x + LOG2_R_DD[idx_x].hi, dx_c0.hi);
+  // The low part is dx^2 * p + low part of (dx * c0) + low part of -log2(r).
+  double log2_x_lo =
+      fputil::multiply_add(dx2, p, dx_c0.lo + LOG2_R_DD[idx_x].lo);
+  // Perform accurate sums.
+  DoubleDouble log2_x = fputil::exact_add(log2_x_hi.hi, log2_x_lo);
+  log2_x.lo += log2_x_hi.lo;
 
   // To compute 2^(y * log2(x)), we break the exponent into 3 parts:
   //   y * log(2) = hi + mid + lo, where