Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 3

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, 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 99.7%

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

4. Taylor expanded in x around inf 51.3%

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

   1. sub-neg 51.3%

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

   2. metadata-eval 51.3%

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

   3. associate-*r/ 51.3%

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

   4. metadata-eval 51.3%

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

6. Simplified 51.3%

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

   1. +-commutative 51.3%

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

   2. distribute-lft-in 51.3%

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

   3. div-inv 51.3%

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

   4. associate-*r* 51.3%

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

   5. pow-flip 51.3%

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

   6. metadata-eval 51.3%

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

8. Applied egg-rr 51.3%

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

   1. fma-define 51.3%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, -0.5, \left(x \cdot 2\right) \cdot {x}^{-2}\right)\right)} \]

   2. associate-*l* 51.3%

      \[\leadsto \log \left(\mathsf{fma}\left(x, -0.5, \color{blue}{x \cdot \left(2 \cdot {x}^{-2}\right)}\right)\right) \]

   3. *-commutative 51.3%

      \[\leadsto \log \left(\mathsf{fma}\left(x, -0.5, \color{blue}{\left(2 \cdot {x}^{-2}\right) \cdot x}\right)\right) \]

   4. associate-*l* 51.3%

      \[\leadsto \log \left(\mathsf{fma}\left(x, -0.5, \color{blue}{2 \cdot \left({x}^{-2} \cdot x\right)}\right)\right) \]

   5. pow-plus 99.7%

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

   6. metadata-eval 99.7%

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

   7. unpow-1 99.7%

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

10. Simplified 99.7%

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, -0.5, 2 \cdot \frac{1}{x}\right)\right)} \]
11. Step-by-step derivation

    1. fma-undefine 99.7%

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

    2. +-commutative 99.7%

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

    3. un-div-inv 99.7%

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

12. Applied egg-rr 99.7%

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

Alternative 3: 99.3% 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. Add Preprocessing

3. Taylor expanded in x around 0 98.9%

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

Reproduce

herbie shell --seed 2024105 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))