Hyperbolic arc-cosine

Average Error: 32.5 → 0.5

Time: 3.2s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))

↓

(FPCore (x) :precision binary64 (+ (log 2.0) (+ (log x) (/ (/ -0.25 x) x))))

C

double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}

↓

double code(double x) {
	return log(2.0) + (log(x) + ((-0.25 / x) / x));
}

Fortran

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(x) + (((-0.25d0) / x) / x))
end function

Java

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(x) + ((-0.25 / x) / x));
}

Python

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

↓

def code(x):
	return math.log(2.0) + (math.log(x) + ((-0.25 / x) / x))

Julia

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

↓

function code(x)
	return Float64(log(2.0) + Float64(log(x) + Float64(Float64(-0.25 / x) / x)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log(2.0) + (log(x) + ((-0.25 / x) / x));
end

Wolfram

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

↓

code[x_] := N[(N[Log[2.0], $MachinePrecision] + N[(N[Log[x], $MachinePrecision] + N[(N[(-0.25 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.5

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

2. Simplified 32.5

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

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. Simplified 0.5

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

5. Final simplification 0.5

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

Reproduce

herbie shell --seed 2022212 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))