Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 6.3s

Alternatives: 6

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: 99.9% 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: 99.9% 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. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2 + -0.5 \cdot {x}^{2}}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ (+ 2.0 (* -0.5 (pow x 2.0))) x)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((2.0d0 + ((-0.5d0) * (x ** 2.0d0))) / x))
end function

Java

public static double code(double x) {
	return Math.log(((2.0 + (-0.5 * Math.pow(x, 2.0))) / x));
}

Python

def code(x):
	return math.log(((2.0 + (-0.5 * math.pow(x, 2.0))) / x))

Julia

function code(x)
	return log(Float64(Float64(2.0 + Float64(-0.5 * (x ^ 2.0))) / x))
end

MATLAB

function tmp = code(x)
	tmp = log(((2.0 + (-0.5 * (x ^ 2.0))) / x));
end

Wolfram

code[x_] := N[Log[N[(N[(2.0 + N[(-0.5 * N[Power[x, 2.0], $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. Taylor expanded in x around 0 99.3%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{fma}\left(x, -0.5, \frac{2}{x}\right) + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (+ (fma x -0.5 (/ 2.0 x)) -1.0)))

C

double code(double x) {
	return log1p((fma(x, -0.5, (2.0 / x)) + -1.0));
}

Julia

function code(x)
	return log1p(Float64(fma(x, -0.5, Float64(2.0 / x)) + -1.0))
end

Wolfram

code[x_] := N[Log[1 + N[(N[(x * -0.5 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + -1.0), $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 99.3%

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

4. Taylor expanded in x around inf 52.4%

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

   1. sub-neg 52.4%

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

   2. associate-*r/ 52.4%

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

   3. metadata-eval 52.4%

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

   4. metadata-eval 52.4%

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

6. Simplified 52.4%

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

7. Taylor expanded in x around inf 52.4%

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

   1. fma-neg 52.4%

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

   2. metadata-eval 52.4%

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

   3. unpow2 52.4%

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

   4. associate-/r* 52.4%

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

   5. unpow-1 52.4%

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

   6. metadata-eval 52.4%

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

   7. pow-plus 52.4%

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

   8. associate-/l* 52.4%

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

   9. *-inverses 52.4%

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

   10. *-rgt-identity 52.4%

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

   11. fma-undefine 52.4%

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

   12. +-commutative 52.4%

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

   13. distribute-lft-out 52.4%

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

   14. fma-define 52.4%

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

   15. *-commutative 52.4%

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

   16. associate-*r* 52.4%

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

   17. pow-plus 99.3%

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

   18. metadata-eval 99.3%

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

   19. unpow-1 99.3%

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

   20. associate-*r/ 99.3%

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

   21. metadata-eval 99.3%

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

9. Simplified 99.3%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, -0.5, \frac{2}{x}\right)\right)} \]
10. Step-by-step derivation

    1. log1p-expm1-u 99.3%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\mathsf{fma}\left(x, -0.5, \frac{2}{x}\right)\right)\right)\right)} \]

    2. expm1-undefine 99.3%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(\mathsf{fma}\left(x, -0.5, \frac{2}{x}\right)\right)} - 1}\right) \]

    3. add-exp-log 99.3%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, -0.5, \frac{2}{x}\right)} - 1\right) \]

11. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, -0.5, \frac{2}{x}\right) - 1\right)} \]

12. Final simplification 99.3%

    \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(x, -0.5, \frac{2}{x}\right) + -1\right) \]
13. Add Preprocessing

Alternative 4: 99.6% 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 99.3%

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

4. Taylor expanded in x around inf 52.4%

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

   1. sub-neg 52.4%

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

   2. associate-*r/ 52.4%

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

   3. metadata-eval 52.4%

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

   4. metadata-eval 52.4%

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

6. Simplified 52.4%

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

7. Taylor expanded in x around inf 52.4%

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

   1. fma-neg 52.4%

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

   2. metadata-eval 52.4%

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

   3. unpow2 52.4%

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

   4. associate-/r* 52.4%

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

   5. unpow-1 52.4%

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

   6. metadata-eval 52.4%

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

   7. pow-plus 52.4%

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

   8. associate-/l* 52.4%

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

   9. *-inverses 52.4%

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

   10. *-rgt-identity 52.4%

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

   11. fma-undefine 52.4%

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

   12. +-commutative 52.4%

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

   13. distribute-lft-out 52.4%

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

   14. fma-define 52.4%

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

   15. *-commutative 52.4%

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

   16. associate-*r* 52.4%

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

   17. pow-plus 99.3%

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

   18. metadata-eval 99.3%

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

   19. unpow-1 99.3%

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

   20. associate-*r/ 99.3%

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

   21. metadata-eval 99.3%

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

9. Simplified 99.3%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, -0.5, \frac{2}{x}\right)\right)} \]
10. Step-by-step derivation

    1. fma-undefine 99.3%

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

    2. +-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

Alternative 5: 99.2% 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 98.3%

   \[\leadsto \color{blue}{\log 2 + -1 \cdot \log x} \]
4. Step-by-step derivation

   1. neg-mul-1 98.3%

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

   2. unsub-neg 98.3%

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

5. Simplified 98.3%

   \[\leadsto \color{blue}{\log 2 - \log x} \]
6. Step-by-step derivation

   1. diff-log 98.7%

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

   2. clear-num 98.7%

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

   3. log-rec 98.7%

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

   4. div-inv 98.7%

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

   5. metadata-eval 98.7%

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

7. Applied egg-rr 98.7%

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

Alternative 6: 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 98.7%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))