Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 4

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. div-inv N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-rgt1-in N/A

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

   6. lift-/.f64 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Log[N[(x * -0.5 + 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. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-neg-in N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

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%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 100.0%

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

Alternative 3: 99.2% 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 99.6

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

5. Applied rewrites 99.6%

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

Alternative 4: 0.0% accurate, 1.4× 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 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 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. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-neg-in N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

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

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%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))