Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 99.5%

Time: 6.3s

Alternatives: 2

Speedup: N/A×

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: 99.5% 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

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(\frac{2 + \frac{-1}{2} \cdot {x}^{2}}{x}\right)}\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(2 + \frac{-1}{2} \cdot {x}^{2}\right), x\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-1}{2} \cdot {x}^{2}\right)\right), x\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left({x}^{2}\right)\right)\right), x\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(x \cdot x\right)\right)\right), x\right)\right) \]

   5. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right) \]

5. Simplified 100.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(\frac{2 + \frac{-1}{2} \cdot {x}^{2}}{x}\right)}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2 + {x}^{2} \cdot \frac{-1}{2}}{x}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2 + {x}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x}\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2 + \left(\mathsf{neg}\left({x}^{2} \cdot \frac{1}{2}\right)\right)}{x}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2 + \left(\mathsf{neg}\left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)}{x}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2 + \left(\mathsf{neg}\left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)}{x}\right)\right) \]

   6. unsub-neg N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2 - x \cdot \left(x \cdot \frac{1}{2}\right)}{x}\right)\right) \]

   7. div-sub N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} - \frac{x \cdot \left(x \cdot \frac{1}{2}\right)}{x}\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} - \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{x}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} - \frac{{x}^{2} \cdot \frac{1}{2}}{x}\right)\right) \]

   10. associate-*l/ N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} - \frac{{x}^{2}}{x} \cdot \frac{1}{2}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} - \frac{x \cdot x}{x} \cdot \frac{1}{2}\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} - \left(x \cdot \frac{x}{x}\right) \cdot \frac{1}{2}\right)\right) \]

   13. *-inverses N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} - \left(x \cdot 1\right) \cdot \frac{1}{2}\right)\right) \]

   14. *-rgt-identity N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} - x \cdot \frac{1}{2}\right)\right) \]

   15. sub-neg N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} + \left(\mathsf{neg}\left(x \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} + x \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} + x \cdot \frac{-1}{2}\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{2}{x} + \frac{-1}{2} \cdot x\right)\right) \]

   19. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\frac{2}{x}\right), \left(\frac{-1}{2} \cdot x\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, x\right), \left(\frac{-1}{2} \cdot x\right)\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, x\right), \left(x \cdot \frac{-1}{2}\right)\right)\right) \]

   22. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(2, x\right), \mathsf{*.f64}\left(x, \frac{-1}{2}\right)\right)\right) \]

8. Simplified 100.0%

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

Alternative 2: 99.1% 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

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(\frac{2}{x}\right)}\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(2, x\right)\right) \]

5. Simplified 99.8%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))