Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.2%

Time: 5.2s

Alternatives: 7

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \log x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* x (* x -0.25)) (- (log 2.0) (log x))))

C

double code(double x) {
	return (x * (x * -0.25)) + (log(2.0) - log(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * (-0.25d0))) + (log(2.0d0) - log(x))
end function

Java

public static double code(double x) {
	return (x * (x * -0.25)) + (Math.log(2.0) - Math.log(x));
}

Python

def code(x):
	return (x * (x * -0.25)) + (math.log(2.0) - math.log(x))

Julia

function code(x)
	return Float64(Float64(x * Float64(x * -0.25)) + Float64(log(2.0) - log(x)))
end

MATLAB

function tmp = code(x)
	tmp = (x * (x * -0.25)) + (log(2.0) - log(x));
end

Wolfram

code[x_] := N[(N[(x * N[(x * -0.25), $MachinePrecision]), $MachinePrecision] + N[(N[Log[2.0], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0 99.7%

   \[\leadsto \color{blue}{\log 2 + \left(-0.25 \cdot {x}^{2} + -1 \cdot \log x\right)} \]
3. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {x}^{2} + -1 \cdot \log x\right) + \log 2} \]

   2. neg-mul-1 99.7%

      \[\leadsto \left(-0.25 \cdot {x}^{2} + \color{blue}{\left(-\log x\right)}\right) + \log 2 \]

   3. associate-+l+ 99.7%

      \[\leadsto \color{blue}{-0.25 \cdot {x}^{2} + \left(\left(-\log x\right) + \log 2\right)} \]

   4. unpow2 99.7%

      \[\leadsto -0.25 \cdot \color{blue}{\left(x \cdot x\right)} + \left(\left(-\log x\right) + \log 2\right) \]

   5. *-commutative 99.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot -0.25} + \left(\left(-\log x\right) + \log 2\right) \]

   6. associate-*l* 99.7%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.25\right)} + \left(\left(-\log x\right) + \log 2\right) \]

   7. +-commutative 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\left(\log 2 + \left(-\log x\right)\right)} \]

   8. unsub-neg 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\left(\log 2 - \log x\right)} \]

4. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \log x\right)} \]

5. Final simplification 99.7%

   \[\leadsto x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \log x\right) \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. expm1-log1p-u 99.6%

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

   2. expm1-udef 99.6%

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

   3. *-un-lft-identity 99.6%

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

   4. div-inv 99.6%

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

   5. distribute-rgt-out 99.6%

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

3. Applied egg-rr 99.6%

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

   1. expm1-def 99.6%

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

   2. expm1-log1p 99.6%

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

   3. associate-*l/ 99.6%

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

   4. +-commutative 99.6%

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

   5. *-lft-identity 99.6%

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

   6. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot -0.5 + 2 \cdot \frac{1}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ (* x -0.5) (* 2.0 (/ 1.0 x)))))

C

double code(double x) {
	return log(((x * -0.5) + (2.0 * (1.0 / x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((x * (-0.5d0)) + (2.0d0 * (1.0d0 / x))))
end function

Java

public static double code(double x) {
	return Math.log(((x * -0.5) + (2.0 * (1.0 / x))));
}

Python

def code(x):
	return math.log(((x * -0.5) + (2.0 * (1.0 / x))))

Julia

function code(x)
	return log(Float64(Float64(x * -0.5) + Float64(2.0 * Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * -0.5) + (2.0 * (1.0 / x))));
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0 99.6%

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

3. Final simplification 99.6%

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

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 99.6%

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

2. Taylor expanded in x around 0 99.2%

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

   1. neg-mul-1 99.2%

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

   2. unsub-neg 99.2%

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

4. Simplified 99.2%

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

   1. diff-log 99.1%

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

   2. clear-num 99.1%

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

   3. log-rec 99.5%

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

   4. div-inv 99.5%

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

   5. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto -\log \left(x \cdot 0.5\right) \]

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 99.6%

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

2. Taylor expanded in x around 0 99.1%

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

3. Final simplification 99.1%

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

Alternative 6: 3.1% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \left(1 + -0.25 \cdot \left(x \cdot x\right)\right) + -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (+ 1.0 (* -0.25 (* x x))) -1.0))

C

double code(double x) {
	return (1.0 + (-0.25 * (x * x))) + -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 + ((-0.25d0) * (x * x))) + (-1.0d0)
end function

Java

public static double code(double x) {
	return (1.0 + (-0.25 * (x * x))) + -1.0;
}

Python

def code(x):
	return (1.0 + (-0.25 * (x * x))) + -1.0

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(-0.25 * Float64(x * x))) + -1.0)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + (-0.25 * (x * x))) + -1.0;
end

Wolfram

code[x_] := N[(N[(1.0 + N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0 99.7%

   \[\leadsto \color{blue}{\log 2 + \left(-0.25 \cdot {x}^{2} + -1 \cdot \log x\right)} \]
3. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {x}^{2} + -1 \cdot \log x\right) + \log 2} \]

   2. neg-mul-1 99.7%

      \[\leadsto \left(-0.25 \cdot {x}^{2} + \color{blue}{\left(-\log x\right)}\right) + \log 2 \]

   3. associate-+l+ 99.7%

      \[\leadsto \color{blue}{-0.25 \cdot {x}^{2} + \left(\left(-\log x\right) + \log 2\right)} \]

   4. unpow2 99.7%

      \[\leadsto -0.25 \cdot \color{blue}{\left(x \cdot x\right)} + \left(\left(-\log x\right) + \log 2\right) \]

   5. *-commutative 99.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot -0.25} + \left(\left(-\log x\right) + \log 2\right) \]

   6. associate-*l* 99.7%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.25\right)} + \left(\left(-\log x\right) + \log 2\right) \]

   7. +-commutative 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\left(\log 2 + \left(-\log x\right)\right)} \]

   8. unsub-neg 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\left(\log 2 - \log x\right)} \]

4. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \log x\right)} \]

5. Taylor expanded in x around inf 99.7%

   \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\left(\log 2 - -1 \cdot \log \left(\frac{1}{x}\right)\right)} \]
6. Step-by-step derivation

   1. log-rec 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - -1 \cdot \color{blue}{\left(-\log x\right)}\right) \]

   2. mul-1-neg 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \color{blue}{\left(-\left(-\log x\right)\right)}\right) \]

   3. remove-double-neg 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \color{blue}{\log x}\right) \]

   4. log-div 99.6%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\log \left(\frac{2}{x}\right)} \]

7. Simplified 99.6%

   \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\log \left(\frac{2}{x}\right)} \]

8. Taylor expanded in x around inf 2.6%

   \[\leadsto \color{blue}{-0.25 \cdot {x}^{2}} \]
9. Step-by-step derivation

   1. unpow2 2.6%

      \[\leadsto -0.25 \cdot \color{blue}{\left(x \cdot x\right)} \]

10. Simplified 2.6%

    \[\leadsto \color{blue}{-0.25 \cdot \left(x \cdot x\right)} \]
11. Step-by-step derivation

    1. expm1-log1p-u 2.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.25 \cdot \left(x \cdot x\right)\right)\right)} \]

    2. expm1-udef 3.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-0.25 \cdot \left(x \cdot x\right)\right)} - 1} \]

    3. log1p-udef 3.1%

       \[\leadsto e^{\color{blue}{\log \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)}} - 1 \]

    4. add-exp-log 3.1%

       \[\leadsto \color{blue}{\left(1 + -0.25 \cdot \left(x \cdot x\right)\right)} - 1 \]

12. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{\left(1 + -0.25 \cdot \left(x \cdot x\right)\right) - 1} \]

13. Final simplification 3.1%

    \[\leadsto \left(1 + -0.25 \cdot \left(x \cdot x\right)\right) + -1 \]

Alternative 7: 2.7% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ -0.25 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* -0.25 (* x x)))

C

double code(double x) {
	return -0.25 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.25d0) * (x * x)
end function

Java

public static double code(double x) {
	return -0.25 * (x * x);
}

Python

def code(x):
	return -0.25 * (x * x)

Julia

function code(x)
	return Float64(-0.25 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = -0.25 * (x * x);
end

Wolfram

code[x_] := N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0 99.7%

   \[\leadsto \color{blue}{\log 2 + \left(-0.25 \cdot {x}^{2} + -1 \cdot \log x\right)} \]
3. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {x}^{2} + -1 \cdot \log x\right) + \log 2} \]

   2. neg-mul-1 99.7%

      \[\leadsto \left(-0.25 \cdot {x}^{2} + \color{blue}{\left(-\log x\right)}\right) + \log 2 \]

   3. associate-+l+ 99.7%

      \[\leadsto \color{blue}{-0.25 \cdot {x}^{2} + \left(\left(-\log x\right) + \log 2\right)} \]

   4. unpow2 99.7%

      \[\leadsto -0.25 \cdot \color{blue}{\left(x \cdot x\right)} + \left(\left(-\log x\right) + \log 2\right) \]

   5. *-commutative 99.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot -0.25} + \left(\left(-\log x\right) + \log 2\right) \]

   6. associate-*l* 99.7%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.25\right)} + \left(\left(-\log x\right) + \log 2\right) \]

   7. +-commutative 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\left(\log 2 + \left(-\log x\right)\right)} \]

   8. unsub-neg 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\left(\log 2 - \log x\right)} \]

4. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \log x\right)} \]

5. Taylor expanded in x around inf 99.7%

   \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\left(\log 2 - -1 \cdot \log \left(\frac{1}{x}\right)\right)} \]
6. Step-by-step derivation

   1. log-rec 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - -1 \cdot \color{blue}{\left(-\log x\right)}\right) \]

   2. mul-1-neg 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \color{blue}{\left(-\left(-\log x\right)\right)}\right) \]

   3. remove-double-neg 99.7%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \left(\log 2 - \color{blue}{\log x}\right) \]

   4. log-div 99.6%

      \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\log \left(\frac{2}{x}\right)} \]

7. Simplified 99.6%

   \[\leadsto x \cdot \left(x \cdot -0.25\right) + \color{blue}{\log \left(\frac{2}{x}\right)} \]

8. Taylor expanded in x around inf 2.6%

   \[\leadsto \color{blue}{-0.25 \cdot {x}^{2}} \]
9. Step-by-step derivation

   1. unpow2 2.6%

      \[\leadsto -0.25 \cdot \color{blue}{\left(x \cdot x\right)} \]

10. Simplified 2.6%

    \[\leadsto \color{blue}{-0.25 \cdot \left(x \cdot x\right)} \]

11. Final simplification 2.6%

    \[\leadsto -0.25 \cdot \left(x \cdot x\right) \]

Reproduce

herbie shell --seed 2023223 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))