Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 5

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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. inv-pow 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. unpow-prod-down 99.6%

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

4. Applied egg-rr 99.6%

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

   1. pow-sqr 99.6%

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

   2. metadata-eval 99.6%

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

6. Simplified 99.6%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0 99.4%

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

   1. *-commutative 99.4%

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

5. Simplified 99.4%

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

   1. pow2 99.4%

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

7. Applied egg-rr 99.4%

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

Alternative 4: 99.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- (log (* x 0.5))))

C

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

Fortran

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((x * 0.5));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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. Add Preprocessing

3. Taylor expanded in x around 0 99.0%

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

   1. clear-num 99.0%

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

   2. log-rec 99.4%

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

   3. div-inv 99.4%

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

   4. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

Alternative 5: 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 99.6%

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

3. Taylor expanded in x around 0 99.0%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))