Average Error: 31.5 → 0.5
Time: 2.9s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\left(\log 2 - \log \left(\frac{1}{x}\right)\right) + 0.25 \cdot \frac{-1}{{x}^{2}} \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (+ (- (log 2.0) (log (/ 1.0 x))) (* 0.25 (/ -1.0 (pow x 2.0)))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return (log(2.0) - log((1.0 / x))) + (0.25 * (-1.0 / pow(x, 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(2.0d0) - log((1.0d0 / x))) + (0.25d0 * ((-1.0d0) / (x ** 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(2.0) - Math.log((1.0 / x))) + (0.25 * (-1.0 / Math.pow(x, 2.0)));
}

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))
↓
def code(x):
	return (math.log(2.0) - math.log((1.0 / x))) + (0.25 * (-1.0 / math.pow(x, 2.0)))

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return Float64(Float64(log(2.0) - log(Float64(1.0 / x))) + Float64(0.25 * Float64(-1.0 / (x ^ 2.0))))
end

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) - 1.0))));
end
↓
function tmp = code(x)
	tmp = (log(2.0) - log((1.0 / x))) + (0.25 * (-1.0 / (x ^ 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[(N[Log[2.0], $MachinePrecision] - N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(-1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\left(\log 2 - \log \left(\frac{1}{x}\right)\right) + 0.25 \cdot \frac{-1}{{x}^{2}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  2. Simplified31.5

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

  3. Taylor expanded in x around inf 0.5

    \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log 2\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]

  4. Final simplification0.5

    \[\leadsto \left(\log 2 - \log \left(\frac{1}{x}\right)\right) + 0.25 \cdot \frac{-1}{{x}^{2}} \]

Reproduce

herbie shell --seed 2022206 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))