Hyperbolic arc-cosine

Average Accuracy: 50.1% → 99.5%

Time: 5.0s

Precision: binary64

Cost: 6848

Math

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

↓

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

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(Float64(x + 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[(N[(x + x), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.1%

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

2. Taylor expanded in x around inf 99.5%

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

3. Simplified 99.5%

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

   [Start] 99.5	\[ \log \left(x + \left(x - 0.5 \cdot \frac{1}{x}\right)\right) \]
   associate-*r/ [=>] 99.5	\[ \log \left(x + \left(x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]
   metadata-eval [=>] 99.5	\[ \log \left(x + \left(x - \frac{\color{blue}{0.5}}{x}\right)\right) \]

4. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ \log \left(x + \left(x - \frac{0.5}{x}\right)\right) \]
   associate-+r- [=>] 99.5	\[ \log \color{blue}{\left(\left(x + x\right) - \frac{0.5}{x}\right)} \]

5. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6848

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

Alternative 2
Accuracy	99.1%
Cost	6592

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

Alternative 3
Accuracy	3.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))