Hyperbolic arc-cosine

Average Error: 32.0 → 0.2

Time: 5.9s

Precision: binary64

Cost: 7360

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return log((((x * 2.0) + (-0.5 / x)) + ((-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((((x * 2.0d0) + ((-0.5d0) / x)) + (((-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((((x * 2.0) + (-0.5 / x)) + ((-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((((x * 2.0) + (-0.5 / x)) + ((-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(Float64(x * 2.0) + Float64(-0.5 / x)) + Float64(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((((x * 2.0) + (-0.5 / x)) + ((-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[(N[(x * 2.0), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 / N[(x * x), $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. Simplified 0.2

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

   [Start] 0.2	\[ \log \left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]
   associate--r+ [=>] 0.2	\[ \log \color{blue}{\left(\left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)} \]
   *-commutative [=>] 0.2	\[ \log \left(\left(\color{blue}{x \cdot 2} - 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right) \]
   associate-*r/ [=>] 0.2	\[ \log \left(\left(x \cdot 2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right) \]
   metadata-eval [=>] 0.2	\[ \log \left(\left(x \cdot 2 - \frac{\color{blue}{0.5}}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right) \]
   associate-*r/ [=>] 0.2	\[ \log \left(\left(x \cdot 2 - \frac{0.5}{x}\right) - \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right) \]
   metadata-eval [=>] 0.2	\[ \log \left(\left(x \cdot 2 - \frac{0.5}{x}\right) - \frac{\color{blue}{0.125}}{{x}^{3}}\right) \]

4. Applied egg-rr 0.2

   \[\leadsto \log \left(\left(x \cdot 2 - \frac{0.5}{x}\right) - \color{blue}{\frac{\frac{0.25}{x \cdot x}}{x} \cdot 0.5}\right) \]

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	6848

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

Alternative 2
Error	0.7
Cost	6592

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

Reproduce

herbie shell --seed 2022356 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))