Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

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, 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. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ (/ 1.0 x) (/ 1.0 x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / 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. Taylor expanded in x around 0 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \log \left(\frac{1}{x} + \frac{1}{x}\right) \]
5. Add Preprocessing

Alternative 3: 31.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ -\log x \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := (-N[Log[x], $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 99.0%

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

4. Taylor expanded in x around 0 98.7%

   \[\leadsto \color{blue}{\log 2 + -1 \cdot \log x} \]

6. Simplified 31.4%

   \[\leadsto \color{blue}{-\log x} \]

7. Final simplification 31.4%

   \[\leadsto -\log x \]
8. Add Preprocessing

Alternative 4: 13.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \log 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return log(2.0)
end

MATLAB

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

Wolfram

code[x_] := N[Log[2.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

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

   3. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.0%

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

6. Taylor expanded in x around 0 98.7%

   \[\leadsto \color{blue}{\log 2 + -1 \cdot \log x} \]

8. Simplified 13.9%

   \[\leadsto \color{blue}{\log 2} \]

9. Final simplification 13.9%

   \[\leadsto \log 2 \]
10. Add Preprocessing

Alternative 5: 3.1% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. pow2 100.0%

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

   3. pow2 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.0%

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

   1. inv-pow 99.0%

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

   2. sqrt-pow1 99.0%

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

   3. metadata-eval 99.0%

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

7. Applied egg-rr 99.0%

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

9. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\log \left(\frac{0}{0}\right) + \log \left(\frac{0}{0}\right)} \]

11. Simplified 3.1%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 3.1%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))