Average Error: 31.9 → 0.7
Time: 2.8s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log x + \log 2 \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary64 (+ (log x) (log 2.0)))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log(x) + log(2.0);
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) - 1.0d0))))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = log(x) + log(2.0d0)
end function

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) - 1.0))));
}
↓
public static double code(double x) {
	return Math.log(x) + Math.log(2.0);
}

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))
↓
def code(x):
	return math.log(x) + math.log(2.0)

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return Float64(log(x) + log(2.0))
end

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - 1.0))));
end
↓
function tmp = code(x)
	tmp = log(x) + log(2.0);
end

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[(N[Log[x], $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log x + \log 2

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

    \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]

  2. Simplified31.9

    \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]

  3. Taylor expanded in x around inf 0.6

    \[\leadsto \log \left(x + \color{blue}{x}\right) \]

  4. Taylor expanded in x around 0 0.7

    \[\leadsto \color{blue}{\log x + \log 2} \]

  5. Final simplification0.7

    \[\leadsto \log x + \log 2 \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))