contract BNum is BConst {
  function btoi(uint256 a) internal pure returns (uint256) {
    return a / BONE;
  }

  function bfloor(uint256 a) internal pure returns (uint256) {
    return btoi(a) * BONE;
  }

  function badd(uint256 a, uint256 b) internal pure returns (uint256) {
    uint256 c = a + b;
    require(c >= a, "ERR_ADD_OVERFLOW");
    return c;
  }

  function bsub(uint256 a, uint256 b) internal pure returns (uint256) {
    (uint256 c, bool flag) = bsubSign(a, b);
    require(!flag, "ERR_SUB_UNDERFLOW");
    return c;
  }

  function bsubSign(uint256 a, uint256 b) internal pure returns (uint256, bool) {
    if (a >= b) {
      return (a - b, false);
    } else {
      return (b - a, true);
    }
  }

  function bmul(uint256 a, uint256 b) internal pure returns (uint256) {
    uint256 c0 = a * b;
    require(a == 0 || c0 / a == b, "ERR_MUL_OVERFLOW");
    uint256 c1 = c0 + (BONE / 2);
    require(c1 >= c0, "ERR_MUL_OVERFLOW");
    uint256 c2 = c1 / BONE;
    return c2;
  }

  function bdiv(uint256 a, uint256 b) internal pure returns (uint256) {
    require(b != 0, "ERR_DIV_ZERO");
    uint256 c0 = a * BONE;
    require(a == 0 || c0 / a == BONE, "ERR_DIV_INTERNAL"); // bmul overflow
    uint256 c1 = c0 + (b / 2);
    require(c1 >= c0, "ERR_DIV_INTERNAL"); //  badd require
    uint256 c2 = c1 / b;
    return c2;
  }

  // DSMath.wpow
  function bpowi(uint256 a, uint256 n) internal pure returns (uint256) {
    uint256 z = n % 2 != 0 ? a : BONE;

    for (n /= 2; n != 0; n /= 2) {
      a = bmul(a, a);

      if (n % 2 != 0) {
        z = bmul(z, a);
      }
    }
    return z;
  }

  // Compute b^(e.w) by splitting it into (b^e)*(b^0.w).
  // Use `bpowi` for `b^e` and `bpowK` for k iterations
  // of approximation of b^0.w
  function bpow(uint256 base, uint256 exp) internal pure returns (uint256) {
    require(base >= MIN_BPOW_BASE, "ERR_BPOW_BASE_TOO_LOW");
    require(base <= MAX_BPOW_BASE, "ERR_BPOW_BASE_TOO_HIGH");

    uint256 whole = bfloor(exp);
    uint256 remain = bsub(exp, whole);

    uint256 wholePow = bpowi(base, btoi(whole));

    if (remain == 0) {
      return wholePow;
    }

    uint256 partialResult = bpowApprox(base, remain, BPOW_PRECISION);
    return bmul(wholePow, partialResult);
  }

  function bpowApprox(
    uint256 base,
    uint256 exp,
    uint256 precision
  ) internal pure returns (uint256) {
    // term 0:
    uint256 a = exp;
    (uint256 x, bool xneg) = bsubSign(base, BONE);
    uint256 term = BONE;
    uint256 sum = term;
    bool negative = false;

    // term(k) = numer / denom
    //         = (product(a - i - 1, i=1-->k) * x^k) / (k!)
    // each iteration, multiply previous term by (a-(k-1)) * x / k
    // continue until term is less than precision
    for (uint256 i = 1; term >= precision; i++) {
      uint256 bigK = i * BONE;
      (uint256 c, bool cneg) = bsubSign(a, bsub(bigK, BONE));
      term = bmul(term, bmul(c, x));
      term = bdiv(term, bigK);
      if (term == 0) break;

      if (xneg) negative = !negative;
      if (cneg) negative = !negative;
      if (negative) {
        sum = bsub(sum, term);
      } else {
        sum = badd(sum, term);
      }
    }

    return sum;
  }
}
function bpow(uint256 base, uint256 exp) internal pure returns (uint256) {
  require(base >= MIN_BPOW_BASE, "ERR_BPOW_BASE_TOO_LOW");
  require(base <= MAX_BPOW_BASE, "ERR_BPOW_BASE_TOO_HIGH");

  uint256 whole = bfloor(exp);
  uint256 remain = bsub(exp, whole);

  uint256 wholePow = bpowi(base, btoi(whole));

  if (remain == 0) {
    return wholePow;
  }

  uint256 partialResult = bpowApprox(base, remain, BPOW_PRECISION);
  return bmul(wholePow, partialResult);
}

// Compute b^(e.w) by splitting it into (b^e)*(b^0.w).
// Use `bpowi` for `b^e` and `bpowK` for k iterations
// of approximation of b^0.w