Average Error: 32.5 → 0.3
Time: 2.9s
Precision: binary64
Cost: 6848
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(x + \left(x + \frac{-0.5}{x}\right)\right) \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary64 (log (+ x (+ x (/ -0.5 x)))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log((x + (x + (-0.5 / x))));
}

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

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))));
}

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))
↓
def code(x):
	return math.log((x + (x + (-0.5 / x))))

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return log(Float64(x + Float64(x + Float64(-0.5 / x))))
end

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

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

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x + \frac{-0.5}{x}\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

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

  2. Taylor expanded in x around inf 0.3

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

  3. Simplified0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x + \frac{-0.5}{x}\right)}\right) \]
    Proof
    (+.f64 x (/.f64 -1/2 x)): 0 points increase in error, 0 points decrease in error
    (+.f64 x (/.f64 (Rewrite<= metadata-eval (neg.f64 1/2)) x)): 0 points increase in error, 0 points decrease in error
    (+.f64 x (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/2 x)))): 0 points increase in error, 0 points decrease in error
    (+.f64 x (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) x))): 0 points increase in error, 0 points decrease in error
    (+.f64 x (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= sub-neg_binary64 (-.f64 x (*.f64 1/2 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error

  4. Final simplification0.3

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

Alternatives

Alternative 1
Error0.6
Cost6592
\[\log \left(x + x\right) \]

Error

Reproduce

herbie shell --seed 2022289 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))