Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

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

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

FPCore

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

C

double code(double x) {
	return log1p((((1.0 + sqrt((1.0 - (x * x)))) / x) + -1.0));
}

Java

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

Python

def code(x):
	return math.log1p((((1.0 + math.sqrt((1.0 - (x * x)))) / x) + -1.0))

Julia

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

Wolfram

code[x_] := N[Log[1 + N[(N[(N[(1.0 + N[Sqrt[N[(1.0 - N[(x * x), $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. +-commutative 100.0%

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

   2. --rgt-identity 100.0%

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

   3. associate--r- 100.0%

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

   4. sqr-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. distribute-neg-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. div-sub 100.0%

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

   9. *-lft-identity 100.0%

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

   10. associate-*l/ 100.0%

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

   11. sub-neg 100.0%

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

   12. sqr-neg 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. log1p-expm1-u 100.0%

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

   2. expm1-undefine 100.0%

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

   3. add-exp-log 100.0%

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

   4. associate-*l/ 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. +-commutative 100.0%

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

   7. 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. Step-by-step derivation

   1. unpow2 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(N[(1.0 + N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $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. +-commutative 100.0%

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

   2. --rgt-identity 100.0%

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

   3. associate--r- 100.0%

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

   4. sqr-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. distribute-neg-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. div-sub 100.0%

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

   9. *-lft-identity 100.0%

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

   10. associate-*l/ 100.0%

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

   11. sub-neg 100.0%

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

   12. sqr-neg 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 99.6% 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. +-commutative 100.0%

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

   2. --rgt-identity 100.0%

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

   3. associate--r- 100.0%

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

   4. sqr-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. distribute-neg-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. div-sub 100.0%

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

   9. *-lft-identity 100.0%

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

   10. associate-*l/ 100.0%

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

   11. sub-neg 100.0%

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

   12. sqr-neg 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

   1. unpow2 100.0%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

   \[\leadsto \log \left(\frac{2 + \left(x \cdot x\right) \cdot -0.5}{x}\right) \]
9. 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. +-commutative 100.0%

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

   2. --rgt-identity 100.0%

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

   3. associate--r- 100.0%

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

   4. sqr-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. distribute-neg-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. div-sub 100.0%

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

   9. *-lft-identity 100.0%

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

   10. associate-*l/ 100.0%

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

   11. sub-neg 100.0%

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

   12. sqr-neg 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 98.9%

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

Alternative 5: 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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. --rgt-identity 100.0%

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

   3. associate--r- 100.0%

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

   4. sqr-neg 100.0%

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

   5. neg-sub0 100.0%

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

   6. distribute-neg-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. div-sub 100.0%

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

   9. *-lft-identity 100.0%

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

   10. associate-*l/ 100.0%

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

   11. sub-neg 100.0%

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

   12. sqr-neg 100.0%

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

   13. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 98.9%

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

   1. clear-num 98.9%

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

   2. log-rec 98.9%

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

   3. div-inv 98.9%

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

   4. metadata-eval 98.9%

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

7. Applied egg-rr 98.9%

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

   1. log1p-expm1-u 5.6%

      \[\leadsto -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x \cdot 0.5\right)\right)\right)} \]

   2. expm1-undefine 5.6%

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

   3. add-exp-log 5.6%

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

9. Applied egg-rr 5.6%

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

10. Taylor expanded in x around 0 3.1%

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

Reproduce

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