Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 5.6s

Alternatives: 4

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: 99.5% 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 99.6%

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

4. Taylor expanded in x around inf 58.0%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 \cdot \frac{1}{{x}^{2}} - 0.5\right)\right)} \]
5. Step-by-step derivation

   1. sub-neg 58.0%

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

   2. associate-*r/ 58.0%

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

   3. metadata-eval 58.0%

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

   4. metadata-eval 58.0%

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

6. Simplified 58.0%

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

   1. clear-num 58.0%

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

   2. inv-pow 58.0%

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

   3. unpow2 58.0%

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

   4. associate-/l* 58.0%

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

   5. unpow-prod-down 58.0%

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

   6. inv-pow 58.0%

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

8. Applied egg-rr 58.0%

   \[\leadsto \log \left(x \cdot \left(\color{blue}{\frac{1}{x} \cdot {\left(\frac{x}{2}\right)}^{-1}} + -0.5\right)\right) \]
9. Step-by-step derivation

   1. unpow-1 58.0%

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

   2. associate-*r/ 58.0%

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

   3. *-rgt-identity 58.0%

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

10. Simplified 58.0%

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

11. Taylor expanded in x around inf 58.0%

    \[\leadsto \log \color{blue}{\left(x \cdot \left(2 \cdot \frac{1}{{x}^{2}} - 0.5\right)\right)} \]
12. Step-by-step derivation

    1. sub-neg 58.0%

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

    2. metadata-eval 58.0%

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

    3. distribute-lft-in 58.0%

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

    4. associate-*r/ 58.0%

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

    5. metadata-eval 58.0%

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

    6. unpow2 58.0%

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

    7. associate-/r* 58.0%

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

    8. associate-*r/ 99.6%

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

    9. *-commutative 99.6%

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

    10. associate-*l/ 99.6%

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

    11. associate-/l* 99.6%

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

    12. *-rgt-identity 99.6%

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

    13. associate-*r/ 99.6%

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

    14. rgt-mult-inverse 99.6%

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

    15. associate-*r/ 99.6%

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

    16. rem-square-sqrt 99.6%

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

    17. associate-*r* 99.6%

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

    18. *-commutative 99.6%

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

    19. *-lft-identity 99.6%

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

    20. rgt-mult-inverse 99.6%

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

    21. associate-*r* 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 3: 99.1% 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 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 98.8%

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

   1. clear-num 98.8%

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

   2. log-div 98.8%

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

   3. metadata-eval 98.8%

      \[\leadsto \color{blue}{0} - \log \left(\frac{x}{2}\right) \]

   4. div-inv 98.8%

      \[\leadsto 0 - \log \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

   5. metadata-eval 98.8%

      \[\leadsto 0 - \log \left(x \cdot \color{blue}{0.5}\right) \]

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{0 - \log \left(x \cdot 0.5\right)} \]
6. Step-by-step derivation

   1. neg-sub0 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 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 98.8%

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

4. Final simplification 98.8%

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

Reproduce

herbie shell --seed 2024130 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))