Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 5

Speedup: 0.6×

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(N[Power[x, -1.0], $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%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x \cdot x, -0.5\right), x, \frac{2}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (fma (fma -0.125 (* x x) -0.5) x (/ 2.0 x))))

C

double code(double x) {
	return log(fma(fma(-0.125, (x * x), -0.5), x, (2.0 / x)));
}

Julia

function code(x)
	return log(fma(fma(-0.125, Float64(x * x), -0.5), x, Float64(2.0 / x)))
end

Wolfram

code[x_] := N[Log[N[(N[(-0.125 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * x + N[(2.0 / 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

   \[\leadsto \log \color{blue}{\left(\frac{2 + {x}^{2} \cdot \left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right)}{x}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \left(\frac{\color{blue}{{x}^{2} \cdot \left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right) + 2}}{x}\right) \]

   2. div-add N/A

      \[\leadsto \log \color{blue}{\left(\frac{{x}^{2} \cdot \left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right)}{x} + \frac{2}{x}\right)} \]

   3. *-commutative N/A

      \[\leadsto \log \left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}}}{x} + \frac{2}{x}\right) \]

   4. associate-/l* N/A

      \[\leadsto \log \left(\color{blue}{\left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \frac{{x}^{2}}{x}} + \frac{2}{x}\right) \]

   5. unpow2 N/A

      \[\leadsto \log \left(\left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \frac{\color{blue}{x \cdot x}}{x} + \frac{2}{x}\right) \]

   6. associate-/l* N/A

      \[\leadsto \log \left(\left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \color{blue}{\left(x \cdot \frac{x}{x}\right)} + \frac{2}{x}\right) \]

   7. *-inverses N/A

      \[\leadsto \log \left(\left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \left(x \cdot \color{blue}{1}\right) + \frac{2}{x}\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \log \left(\left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \color{blue}{x} + \frac{2}{x}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}, x, \frac{2}{x}\right)\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}}, x, \frac{2}{x}\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot {x}^{2}} - \frac{1}{2}, x, \frac{2}{x}\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{2}, x, \frac{2}{x}\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(\frac{-1}{8} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{2}, x, \frac{2}{x}\right)\right) \]

   14. lower-/.f64 99.8

       \[\leadsto \log \left(\mathsf{fma}\left(-0.125 \cdot \left(x \cdot x\right) - 0.5, x, \color{blue}{\frac{2}{x}}\right)\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(-0.125 \cdot \left(x \cdot x\right) - 0.5, x, \frac{2}{x}\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \log \left(\mathsf{fma}\left(\frac{-1}{8} \cdot {x}^{2} - \frac{1}{2}, x, \frac{2}{x}\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.8%

   \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x \cdot x, -0.5\right), x, \frac{2}{x}\right)\right) \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(-0.5, x, \frac{2}{x}\right)\right) \end{array} \]

FPCore

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

C

double code(double x) {
	return log(fma(-0.5, x, (2.0 / x)));
}

Julia

function code(x)
	return log(fma(-0.5, x, Float64(2.0 / x)))
end

Wolfram

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

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

   1. +-commutative N/A

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

   2. div-add N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(-0.5, x, \frac{2}{x}\right)\right)} \]
6. Add Preprocessing

Alternative 4: 99.1% accurate, 1.3× 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

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

   1. lower-/.f64 99.0

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

5. Applied rewrites 99.0%

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

Alternative 5: 0.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(-0.5 * 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

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

   1. +-commutative N/A

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

   2. div-add N/A

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

   3. associate-/l* N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 0.0%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))