Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 100.0%

Time: 9.6s

Alternatives: 7

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: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log((x / (1.0d0 + sqrt((1.0d0 - (x ** 2.0d0))))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = -log((x / (1.0 + sqrt((1.0 - (x ^ 2.0))))));
end

Wolfram

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

   1. add-sqr-sqrt 100.0%

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

   2. log-prod 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. div-inv 100.0%

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

   5. distribute-rgt-out 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. div-inv 100.0%

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

   9. distribute-rgt-out 100.0%

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

4. Applied egg-rr 100.0%

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

   1. count-2 100.0%

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

   2. associate-*l/ 100.0%

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

   3. *-lft-identity 100.0%

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

6. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. add-log-exp 100.0%

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

   3. exp-to-pow 100.0%

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

   4. pow2 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. clear-num 100.0%

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

   7. log-div 100.0%

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

   8. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 - \log \left(\frac{x}{1 + \sqrt{1 - {x}^{2}}}\right)} \]
9. Step-by-step derivation

   1. neg-sub0 100.0%

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

10. Simplified 100.0%

    \[\leadsto \color{blue}{-\log \left(\frac{x}{1 + \sqrt{1 - {x}^{2}}}\right)} \]
11. Add Preprocessing

Alternative 2: 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 3: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(N[(2.0 + N[(N[(x * x), $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. Add Preprocessing

3. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

5. Simplified 99.5%

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

   1. unpow2 99.5%

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

7. Applied egg-rr 99.5%

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

Alternative 4: 99.0% 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 98.9%

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

   3. metadata-eval 98.9%

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

   4. div-inv 98.9%

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

   5. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

   1. neg-sub0 98.9%

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

7. Simplified 98.9%

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

Alternative 5: 99.0% 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. Add Preprocessing

Alternative 6: 3.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ -\mathsf{log1p}\left(-1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (log1p -1.0)))

C

double code(double x) {
	return -log1p(-1.0);
}

Java

public static double code(double x) {
	return -Math.log1p(-1.0);
}

Python

def code(x):
	return -math.log1p(-1.0)

Julia

function code(x)
	return Float64(-log1p(-1.0))
end

Wolfram

code[x_] := (-N[Log[1 + -1.0], $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 98.9%

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

   3. metadata-eval 98.9%

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

   4. div-inv 98.9%

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

   5. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

   1. neg-sub0 98.9%

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

7. Simplified 98.9%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 1.5%

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

   3. sqr-neg 1.5%

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

   4. sqrt-unprod 1.5%

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

   5. add-sqr-sqrt 1.5%

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

   6. neg-log 1.5%

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

   7. *-commutative 1.5%

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

   8. associate-/r* 1.5%

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

   9. metadata-eval 1.5%

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

   10. log1p-expm1-u 1.5%

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

   11. expm1-define 1.5%

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

   12. add-exp-log 1.5%

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

   13. sub-neg 1.5%

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

9. Applied egg-rr 6.3%

   \[\leadsto -\color{blue}{\mathsf{log1p}\left(x \cdot 0.5 + -1\right)} \]

10. Taylor expanded in x around 0 3.1%

    \[\leadsto -\mathsf{log1p}\left(\color{blue}{-1}\right) \]
11. Add Preprocessing

Alternative 7: 3.1% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

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 98.9%

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

   3. metadata-eval 98.9%

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

   4. div-inv 98.9%

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

   5. metadata-eval 98.9%

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

5. Applied egg-rr 98.9%

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

   1. neg-sub0 98.9%

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

7. Simplified 98.9%

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

9. Applied egg-rr 3.1%

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

    1. *-inverses 3.1%

       \[\leadsto -\log \color{blue}{1} \]

11. Simplified 3.1%

    \[\leadsto -\log \color{blue}{1} \]
12. Step-by-step derivation

    1. metadata-eval 3.1%

       \[\leadsto -\color{blue}{0} \]

13. Applied egg-rr 3.1%

    \[\leadsto -\color{blue}{0} \]

14. Final simplification 3.1%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))