Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (log (/ (+ 1.0 (sqrt (- 1.0 (* x 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 + sqrt((1.0 - (x * 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 + sqrt((1.0d0 - (x * 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 + Math.sqrt((1.0 - (x * 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 + math.sqrt((1.0 - (x * 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(Float64(1.0 + sqrt(Float64(1.0 - Float64(x * 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 + sqrt((1.0 - (x * 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[(N[(1.0 + N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
   expm1-log1p-u [=>] 100.0	\[ \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\right)\right)} \]
   expm1-udef [=>] 100.0	\[ \log \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)} - 1\right)} \]
   +-commutative [=>] 100.0	\[ \log \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1 - x \cdot x}}{x} + \frac{1}{x}}\right)} - 1\right) \]
   inv-pow [=>] 100.0	\[ \log \left(e^{\mathsf{log1p}\left(\frac{\sqrt{1 - x \cdot x}}{x} + \color{blue}{{x}^{-1}}\right)} - 1\right) \]

3. Simplified 100.0%

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

   [Start] 100.0	\[ \log \left(e^{\mathsf{log1p}\left(\frac{\sqrt{1 - x \cdot x}}{x} + {x}^{-1}\right)} - 1\right) \]
   expm1-def [=>] 100.0	\[ \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{1 - x \cdot x}}{x} + {x}^{-1}\right)\right)\right)} \]
   expm1-log1p [=>] 100.0	\[ \log \color{blue}{\left(\frac{\sqrt{1 - x \cdot x}}{x} + {x}^{-1}\right)} \]
   *-rgt-identity [<=] 100.0	\[ \log \left(\frac{\color{blue}{\sqrt{1 - x \cdot x} \cdot 1}}{x} + {x}^{-1}\right) \]
   associate-*r/ [<=] 100.0	\[ \log \left(\color{blue}{\sqrt{1 - x \cdot x} \cdot \frac{1}{x}} + {x}^{-1}\right) \]
   unpow-1 [<=] 100.0	\[ \log \left(\sqrt{1 - x \cdot x} \cdot \color{blue}{{x}^{-1}} + {x}^{-1}\right) \]
   distribute-lft1-in [=>] 100.0	\[ \log \color{blue}{\left(\left(\sqrt{1 - x \cdot x} + 1\right) \cdot {x}^{-1}\right)} \]
   unpow-1 [=>] 100.0	\[ \log \left(\left(\sqrt{1 - x \cdot x} + 1\right) \cdot \color{blue}{\frac{1}{x}}\right) \]
   associate-*r/ [=>] 100.0	\[ \log \color{blue}{\left(\frac{\left(\sqrt{1 - x \cdot x} + 1\right) \cdot 1}{x}\right)} \]
   distribute-rgt1-in [<=] 100.0	\[ \log \left(\frac{\color{blue}{1 + \sqrt{1 - x \cdot x} \cdot 1}}{x}\right) \]
   *-rgt-identity [=>] 100.0	\[ \log \left(\frac{1 + \color{blue}{\sqrt{1 - x \cdot x}}}{x}\right) \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6976

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

Alternative 2
Accuracy	99.5%
Cost	6976

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

Alternative 3
Accuracy	99.2%
Cost	6656

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

Alternative 4
Accuracy	99.1%
Cost	6592

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

Reproduce

herbie shell --seed 2023159 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))