Hyperbolic arc-(co)secant

Average Accuracy: 100.0% → 99.3%

Time: 5.4s

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 99.6%

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

2. Taylor expanded in x around 0 99.3%

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

3. Applied egg-rr 99.7%

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

   [Start] 99.3	\[ \log \left(\frac{2}{x}\right) \]
   clear-num [=>] 99.3	\[ \log \color{blue}{\left(\frac{1}{\frac{x}{2}}\right)} \]
   log-div [=>] 99.7	\[ \color{blue}{\log 1 - \log \left(\frac{x}{2}\right)} \]
   metadata-eval [=>] 99.7	\[ \color{blue}{0} - \log \left(\frac{x}{2}\right) \]
   div-inv [=>] 99.7	\[ 0 - \log \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]
   metadata-eval [=>] 99.7	\[ 0 - \log \left(x \cdot \color{blue}{0.5}\right) \]

4. Simplified 99.7%

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

   [Start] 99.7	\[ 0 - \log \left(x \cdot 0.5\right) \]
   neg-sub0 [<=] 99.7	\[ \color{blue}{-\log \left(x \cdot 0.5\right)} \]

5. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	6592

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

Reproduce

herbie shell --seed 2023157 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))