Hyperbolic arc-cosine

Average Accuracy: 49.7% → 98.7%

Time: 4.9s

Precision: binary64

Cost: 12992

Math

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

↓

\[\log 2 + \log x \]

FPCore

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

↓

(FPCore (x) :precision binary64 (+ (log 2.0) (log x)))

C

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

↓

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

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)

Julia

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

↓

function code(x)
	return Float64(log(2.0) + log(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);
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[Log[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in x around inf 98.7%

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

3. Simplified 98.7%

   \[\leadsto \color{blue}{\log 2 + \log x} \]
   Proof

   [Start] 98.7	\[ \log 2 + -1 \cdot \log \left(\frac{1}{x}\right) \]
   mul-1-neg [=>] 98.7	\[ \log 2 + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} \]
   log-rec [=>] 98.7	\[ \log 2 + \left(-\color{blue}{\left(-\log x\right)}\right) \]
   remove-double-neg [=>] 98.7	\[ \log 2 + \color{blue}{\log x} \]

4. Final simplification 98.7%

   \[\leadsto \log 2 + \log x \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	6848

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

Alternative 2
Accuracy	98.9%
Cost	6592

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

Alternative 3
Accuracy	3.1%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))