Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 4.2s

Alternatives: 5

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, 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 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. +-commutative 100.0%

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

   4. div-inv 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. cancel-sign-sub-inv 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-lft-identity 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-def 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-rgt-neg-out 100.0%

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

   9. unsub-neg 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := (-N[Log[N[(x / N[(-0.5 * N[(x * x), $MachinePrecision] + 2.0), $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. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. +-commutative 100.0%

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

   4. div-inv 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. cancel-sign-sub-inv 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-lft-identity 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-def 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-rgt-neg-out 100.0%

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

   9. unsub-neg 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 98.9%

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

   1. unpow2 98.9%

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

8. Simplified 98.9%

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

   1. clear-num 98.9%

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

   2. log-rec 98.9%

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

   3. +-commutative 98.9%

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

   4. fma-def 98.9%

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

10. Applied egg-rr 98.9%

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

11. Final simplification 98.9%

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

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 100.0%

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

2. Taylor expanded in x around 0 98.9%

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

3. Final simplification 98.9%

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

Alternative 4: 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. Step-by-step derivation

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. +-commutative 100.0%

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

   4. div-inv 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. cancel-sign-sub-inv 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-lft-identity 100.0%

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

   5. +-commutative 100.0%

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

   6. fma-def 100.0%

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

   7. +-commutative 100.0%

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

   8. distribute-rgt-neg-out 100.0%

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

   9. unsub-neg 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 98.9%

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

   1. unpow2 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 5: 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. Taylor expanded in x around 0 98.2%

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

3. Final simplification 98.2%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))