Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.1

Time: 6.1s

Precision: binary64

Cost: 13504

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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((1.0d0 / (x / (1.0d0 + sqrt((1.0d0 - (x * x)))))))
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((1.0 / (x / (1.0 + Math.sqrt((1.0 - (x * x)))))));
}

Python

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

↓

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

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

MATLAB

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

↓

function tmp = code(x)
	tmp = log((1.0 / (x / (1.0 + sqrt((1.0 - (x * x)))))));
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[(1.0 / N[(x / N[(1.0 + N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

   \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)} - 1\right)} \]

3. Simplified 0.1

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

   [Start] 0.1	\[ \log \left(e^{\mathsf{log1p}\left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)} - 1\right) \]
   expm1-def [=>] 0.1	\[ \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\right)\right)} \]
   expm1-log1p [=>] 0.1	\[ \log \color{blue}{\left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)} \]
   *-lft-identity [<=] 0.1	\[ \log \color{blue}{\left(1 \cdot \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\right)} \]
   distribute-lft-in [=>] 0.1	\[ \log \color{blue}{\left(1 \cdot \frac{1}{x} + 1 \cdot \frac{\sqrt{1 - x \cdot x}}{x}\right)} \]
   associate-*r/ [=>] 0.1	\[ \log \left(1 \cdot \frac{1}{x} + \color{blue}{\frac{1 \cdot \sqrt{1 - x \cdot x}}{x}}\right) \]
   *-commutative [=>] 0.1	\[ \log \left(1 \cdot \frac{1}{x} + \frac{\color{blue}{\sqrt{1 - x \cdot x} \cdot 1}}{x}\right) \]
   associate-*r/ [<=] 0.1	\[ \log \left(1 \cdot \frac{1}{x} + \color{blue}{\sqrt{1 - x \cdot x} \cdot \frac{1}{x}}\right) \]
   distribute-rgt-in [<=] 0.1	\[ \log \color{blue}{\left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - x \cdot x}\right)\right)} \]
   associate-*l/ [=>] 0.1	\[ \log \color{blue}{\left(\frac{1 \cdot \left(1 + \sqrt{1 - x \cdot x}\right)}{x}\right)} \]
   *-lft-identity [=>] 0.1	\[ \log \left(\frac{\color{blue}{1 + \sqrt{1 - x \cdot x}}}{x}\right) \]

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13376

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

Alternative 2
Error	0.3
Cost	7104

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

Alternative 3
Error	0.3
Cost	6976

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

Alternative 4
Error	0.6
Cost	6592

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

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))