Hyperbolic arc-cosine

Average Error: 31.5 → 0.1

Time: 2.9s

Precision: binary64

Cost: 19776

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 + (sqrt((x + 1.0d0)) * sqrt((x + (-1.0d0))))))
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 + (Math.sqrt((x + 1.0)) * Math.sqrt((x + -1.0)))));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x)
	tmp = log((x + (sqrt((x + 1.0)) * sqrt((x + -1.0)))));
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[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 31.5

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

2. Applied egg-rr 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.4
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 2023076 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))