Hyperbolic arc-cosine

Average Error: 31.9 → 0.3

Time: 3.5s

Precision: binary64

Cost: 6848

Math

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

↓

\[\log \left(x + \left(x + \frac{-0.5}{x}\right)\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (log (+ x (+ x (/ -0.5 x)))))

C

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

↓

double code(double x) {
	return log((x + (x + (-0.5 / 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((x + (x + ((-0.5d0) / 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((x + (x + (-0.5 / x))));
}

Python

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

↓

def code(x):
	return math.log((x + (x + (-0.5 / x))))

Julia

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

↓

function code(x)
	return log(Float64(x + Float64(x + Float64(-0.5 / x))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log((x + (x + (-0.5 / 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[Log[N[(x + N[(x + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 31.9

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

2. Taylor expanded in x around inf 0.3

   \[\leadsto \log \left(x + \color{blue}{\left(x - 0.5 \cdot \frac{1}{x}\right)}\right) \]

3. Simplified 0.3

   \[\leadsto \log \left(x + \color{blue}{\left(x - \frac{0.5}{x}\right)}\right) \]
   Proof

   [Start] 0.3	\[ \log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]
   associate-*r/ [=>] 0.3	\[ \log \left(x + \left(x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]
   metadata-eval [=>] 0.3	\[ \log \left(x + \left(x - \frac{\color{blue}{0.5}}{x}\right)\right) \]

4. Final simplification 0.3

   \[\leadsto \log \left(x + \left(x + \frac{-0.5}{x}\right)\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	6592

\[\log \left(x + x\right) \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))