Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.6

Time: 4.3s

Precision: binary64

Cost: 6656

Math

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

↓

\[-\log \left(x \cdot 0.5\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (- (log (* x 0.5))))

C

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

↓

double code(double x) {
	return -log((x * 0.5));
}

Fortran

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((x * 0.5d0))
end function

Java

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((x * 0.5));
}

Python

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

↓

def code(x):
	return -math.log((x * 0.5))

Julia

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

↓

function code(x)
	return Float64(-log(Float64(x * 0.5)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = -log((x * 0.5));
end

Wolfram

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[(x * 0.5), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.6

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

3. Applied egg-rr 0.6

   \[\leadsto \color{blue}{0 - \log \left(x \cdot 0.5\right)} \]

4. Simplified 0.6

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

   [Start] 0.6	\[ 0 - \log \left(x \cdot 0.5\right) \]
   sub0-neg [=>] 0.6	\[ \color{blue}{-\log \left(x \cdot 0.5\right)} \]

5. Final simplification 0.6

   \[\leadsto -\log \left(x \cdot 0.5\right) \]

Alternatives

Alternative 1
Error	0.6
Cost	6592

\[\log \left(\frac{2}{x}\right) \]

Reproduce

herbie shell --seed 2023027 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))