?

Average Accuracy: 100.0% → 99.6%
Time: 2.4s
Precision: binary64
Cost: 6976

?

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

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((1.0d0 / x) + (sqrt((1.0d0 - (x * x))) / x)))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = log((((-0.5d0) * x) + (2.0d0 * (1.0d0 / x))))
end function

public static double code(double x) {
	return Math.log(((1.0 / x) + (Math.sqrt((1.0 - (x * x))) / x)));
}
↓
public static double code(double x) {
	return Math.log(((-0.5 * x) + (2.0 * (1.0 / x))));
}

def code(x):
	return math.log(((1.0 / x) + (math.sqrt((1.0 - (x * x))) / x)))
↓
def code(x):
	return math.log(((-0.5 * x) + (2.0 * (1.0 / x))))

function code(x)
	return log(Float64(Float64(1.0 / x) + Float64(sqrt(Float64(1.0 - Float64(x * x))) / x)))
end
↓
function code(x)
	return log(Float64(Float64(-0.5 * x) + Float64(2.0 * Float64(1.0 / x))))
end

function tmp = code(x)
	tmp = log(((1.0 / x) + (sqrt((1.0 - (x * x))) / x)));
end
↓
function tmp = code(x)
	tmp = log(((-0.5 * x) + (2.0 * (1.0 / x))));
end

code[x_] := N[Log[N[(N[(1.0 / x), $MachinePrecision] + N[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[Log[N[(N[(-0.5 * x), $MachinePrecision] + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

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

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 100.0%

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

  2. Taylor expanded in x around 0 99.6%

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

  3. Final simplification99.6%

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

Alternatives

Alternative 1
Accuracy99.1%
Cost6592
\[\log \left(\frac{2}{x}\right) \]

Error

Reproduce?

herbie shell --seed 2023146 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))