Function Acosh

tensorflow/compiler/xla/client/lib/math.cc:1146–1174

acosh(x) = log(x + sqrt(x^2 - 1)) if x >= -1 = log(x + sqrt((x+1)*(x-1))) acosh(x) = nan if x < -1 If x^2 will overflow, we approximate sqrt(x^2 - 1) == x and compute as log(2*x) = log(2) + log(x). (Note this works because negative x never overflows; x < -1 simply yields nan. This is quite different than asinh!)

// log(2*x) = log(2) + log(x).  (Note this works because negative x never
// overflows; x < -1 simply yields nan.  This is quite different than asinh!)
XlaOp Acosh(XlaOp x) {
  XlaBuilder* b = x.builder();
  return b->ReportErrorOrReturn([&]() -> StatusOr<XlaOp> {
    TF_ASSIGN_OR_RETURN(auto shape, b->GetShape(x));

    auto one = ScalarLike(x, 1);
    auto neg_one = ScalarLike(x, -1);
    auto nan = FullLike(x, std::numeric_limits<float>::quiet_NaN());

    // return
    //
    //   nan                        if x < -1
    //   log(x) + log(2)            if x >= sqrt_max_value
    //   log(x + sqrt((x+1)*(x-1))) otherwise
    //
    // TODO(jlebar): For now, we ignore the question of overflow if x is a
    // complex type, because we don't yet have exhaustive tests for complex trig
    // functions.
    auto naive_result = Log(x + Sqrt((x + one) * (x - one)));
    if (primitive_util::IsComplexType(shape.element_type())) {
      return naive_result;
    }
    auto overflow_result = Log(x) + Log(ScalarLike(x, 2));

    auto sqrt_max_value = Sqrt(MaxFiniteValue(b, shape.element_type()));
    return Select(Lt(x, neg_one), nan,
                  Select(Ge(x, sqrt_max_value), overflow_result, naive_result));
  });
}

// asinh(x) = log(x + sqrt(x^2 + 1))
//