Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x) {
	return log(((1.0 / x) + (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

Java

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

Python

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

MATLAB

function tmp = code(x)
	tmp = log(((1.0 / x) + (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]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return log(((1.0 / x) + (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

Java

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

Python

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

MATLAB

function tmp = code(x)
	tmp = log(((1.0 / x) + (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]

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1}{x} + \frac{\frac{1}{x}}{{\left(1 - {x}^{2}\right)}^{-0.5}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (+ (/ 1.0 x) (/ (/ 1.0 x) (pow (- 1.0 (pow x 2.0)) -0.5)))))

C

double code(double x) {
	return log(((1.0 / x) + ((1.0 / x) / pow((1.0 - pow(x, 2.0)), -0.5))));
}

Fortran

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

Java

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

Python

def code(x):
	return math.log(((1.0 / x) + ((1.0 / x) / math.pow((1.0 - math.pow(x, 2.0)), -0.5))))

Julia

function code(x)
	return log(Float64(Float64(1.0 / x) + Float64(Float64(1.0 / x) / (Float64(1.0 - (x ^ 2.0)) ^ -0.5))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

   3. div-inv 100.0%

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

   4. unpow-prod-down 100.0%

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

   5. inv-pow 100.0%

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

   6. pow1/2 100.0%

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

   7. pow-flip 100.0%

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

   8. pow2 100.0%

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

   9. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

   \[\leadsto \log \left(\frac{1}{x} + \color{blue}{\frac{1}{x} \cdot {\left({\left(1 - {x}^{2}\right)}^{-0.5}\right)}^{-1}}\right) \]
4. Step-by-step derivation

   1. unpow-1 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-rgt-identity 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return log(((1.0 / x) + (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

Java

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

Python

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

MATLAB

function tmp = code(x)
	tmp = log(((1.0 / x) + (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]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ (* x -0.5) (* (/ 1.0 x) 2.0))))

C

double code(double x) {
	return log(((x * -0.5) + ((1.0 / x) * 2.0)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 99.7%

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

3. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ 2.0 x)))

C

double code(double x) {
	return log((2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((2.0d0 / x))
end function

Java

public static double code(double x) {
	return Math.log((2.0 / x));
}

Python

def code(x):
	return math.log((2.0 / x))

Julia

function code(x)
	return log(Float64(2.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = log((2.0 / x));
end

Wolfram

code[x_] := N[Log[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]

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 99.7%

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

3. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.4%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))