Hyperbolic arc-cosine

?

Percentage Accurate: 51.4% → 99.5%
Time: 5.5s
Precision: binary64
Cost: 6848

?

\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(x \cdot 2 - \frac{0.5}{x}\right) \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary64 (log (- (* x 2.0) (/ 0.5 x))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log(((x * 2.0) - (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 * 2.0d0) - (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 * 2.0) - (0.5 / x)));
}

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)))

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return log(Float64(Float64(x * 2.0) - 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 * 2.0) - (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[(N[(x * 2.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x \cdot 2 - \frac{0.5}{x}\right)

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 54.6%

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

  2. Taylor expanded in x around inf 99.4%

    \[\leadsto \log \color{blue}{\left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right)} \]

  3. Simplified99.4%

    \[\leadsto \log \color{blue}{\left(x \cdot 2 - \frac{0.5}{x}\right)} \]
    Step-by-step derivation

    [Start]99.4

    \[ \log \left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right) \]

    *-commutative [=>]99.4

    \[ \log \left(\color{blue}{x \cdot 2} - 0.5 \cdot \frac{1}{x}\right) \]

    associate-*r/ [=>]99.4

    \[ \log \left(x \cdot 2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) \]

    metadata-eval [=>]99.4

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

  4. Final simplification99.4%

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

Alternatives

Alternative 1
Accuracy99.1%
Cost6592
\[\log \left(x + x\right) \]
Alternative 2
Accuracy3.1%
Cost64
\[0 \]

Error

Reproduce?

herbie shell --seed 2023160 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))