Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 3.5s

Alternatives: 4

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: 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, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (log1p (+ (/ (+ 1.0 (sqrt (- 1.0 (pow x 2.0)))) x) -1.0)))

C

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

Java

public static double code(double x) {
	return Math.log1p((((1.0 + Math.sqrt((1.0 - Math.pow(x, 2.0)))) / x) + -1.0));
}

Python

def code(x):
	return math.log1p((((1.0 + math.sqrt((1.0 - math.pow(x, 2.0)))) / x) + -1.0))

Julia

function code(x)
	return log1p(Float64(Float64(Float64(1.0 + sqrt(Float64(1.0 - (x ^ 2.0)))) / x) + -1.0))
end

Wolfram

code[x_] := N[Log[1 + N[(N[(N[(1.0 + N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x), $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. Step-by-step derivation

   1. sqr-neg 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg2 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/r* 100.0%

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

   7. div-sub 100.0%

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

   8. *-lft-identity 100.0%

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

   9. associate-*l/ 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-neg-frac2 100.0%

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

   12. metadata-eval 100.0%

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

   13. /-rgt-identity 100.0%

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

3. Simplified 100.0%

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

   1. log1p-expm1-u 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - x \cdot x}\right)\right)\right)\right)} \]

   2. expm1-undefine 100.0%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - x \cdot x}\right)\right)} - 1}\right) \]

   3. add-exp-log 100.0%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1}{x} \cdot \left(1 + \sqrt{1 - x \cdot x}\right)} - 1\right) \]

   4. associate-*l/ 100.0%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(1 + \sqrt{1 - x \cdot x}\right)}{x}} - 1\right) \]

   5. *-un-lft-identity 100.0%

      \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{1 + \sqrt{1 - x \cdot x}}}{x} - 1\right) \]

   6. pow2 100.0%

      \[\leadsto \mathsf{log1p}\left(\frac{1 + \sqrt{1 - \color{blue}{{x}^{2}}}}{x} - 1\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1 + \sqrt{1 - {x}^{2}}}{x} - 1\right)} \]

7. Final simplification 100.0%

   \[\leadsto \mathsf{log1p}\left(\frac{1 + \sqrt{1 - {x}^{2}}}{x} + -1\right) \]
8. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right) \]
2. Step-by-step derivation

   1. sqr-neg 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg2 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/r* 100.0%

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

   7. div-sub 100.0%

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

   8. *-lft-identity 100.0%

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

   9. associate-*l/ 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-neg-frac2 100.0%

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

   12. metadata-eval 100.0%

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

   13. /-rgt-identity 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(N[(2.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $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. Step-by-step derivation

   1. sqr-neg 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg2 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/r* 100.0%

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

   7. div-sub 100.0%

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

   8. *-lft-identity 100.0%

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

   9. associate-*l/ 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-neg-frac2 100.0%

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

   12. metadata-eval 100.0%

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

   13. /-rgt-identity 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.2% 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. Step-by-step derivation

   1. sqr-neg 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-frac-neg2 100.0%

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

   4. sub-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/r* 100.0%

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

   7. div-sub 100.0%

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

   8. *-lft-identity 100.0%

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

   9. associate-*l/ 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-neg-frac2 100.0%

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

   12. metadata-eval 100.0%

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

   13. /-rgt-identity 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 98.9%

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

6. Final simplification 98.9%

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

Reproduce

herbie shell --seed 2024112 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))