Hyperbolic arc-cosine

Average Error: 32.0 → 0.2

Time: 3.9s

Precision: binary64

Cost: 7232

Math

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

↓

\[\log \left(2 \cdot x + \frac{-0.5 - \frac{0.125}{x \cdot x}}{x}\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (log (+ (* 2.0 x) (/ (- -0.5 (/ 0.125 (* x x))) x))))

C

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

↓

double code(double x) {
	return log(((2.0 * x) + ((-0.5 - (0.125 / (x * 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 * x) + (((-0.5d0) - (0.125d0 / (x * 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 * x) + ((-0.5 - (0.125 / (x * x))) / x)));
}

Python

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

↓

def code(x):
	return math.log(((2.0 * x) + ((-0.5 - (0.125 / (x * x))) / x)))

Julia

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

↓

function code(x)
	return log(Float64(Float64(2.0 * x) + Float64(Float64(-0.5 - Float64(0.125 / Float64(x * 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 * x) + ((-0.5 - (0.125 / (x * 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[Log[N[(N[(2.0 * x), $MachinePrecision] + N[(N[(-0.5 - N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 32.0

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

2. Taylor expanded in x around inf 0.2

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

3. Applied egg-rr 0.2

   \[\leadsto \log \left(2 \cdot x - \color{blue}{\frac{1}{x} \cdot \left(0.5 + \frac{0.125}{x \cdot x}\right)}\right) \]

4. Simplified 0.2

   \[\leadsto \log \left(2 \cdot x - \color{blue}{\frac{0.5 + \frac{0.125}{x \cdot x}}{x}}\right) \]
   Proof

   [Start] 0.2	\[ \log \left(2 \cdot x - \frac{1}{x} \cdot \left(0.5 + \frac{0.125}{x \cdot x}\right)\right) \]
   associate-*l/ [=>] 0.2	\[ \log \left(2 \cdot x - \color{blue}{\frac{1 \cdot \left(0.5 + \frac{0.125}{x \cdot x}\right)}{x}}\right) \]
   *-lft-identity [=>] 0.2	\[ \log \left(2 \cdot x - \frac{\color{blue}{0.5 + \frac{0.125}{x \cdot x}}}{x}\right) \]

5. Final simplification 0.2

   \[\leadsto \log \left(2 \cdot x + \frac{-0.5 - \frac{0.125}{x \cdot x}}{x}\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	6848

\[\log \left(2 \cdot x + \frac{-0.5}{x}\right) \]

Alternative 2
Error	0.6
Cost	6592

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

Reproduce

herbie shell --seed 2023041 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))