Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

Alternatives: 6

Speedup: 1.1×

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, 1.1× 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. frac-2neg N/A

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

   5. neg-mul-1 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. frac-2neg N/A

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

   11. lift-/.f64 N/A

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

   12. distribute-rgt1-in N/A

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

   13. lift-/.f64 N/A

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

   14. un-div-inv N/A

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

   15. lower-/.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ -\log \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0625, 0.125\right), 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (log (* x (fma (* x x) (fma x (* x 0.0625) 0.125) 0.5)))))

C

double code(double x) {
	return -log((x * fma((x * x), fma(x, (x * 0.0625), 0.125), 0.5)));
}

Julia

function code(x)
	return Float64(-log(Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.0625), 0.125), 0.5))))
end

Wolfram

code[x_] := (-N[Log[N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0625), $MachinePrecision] + 0.125), $MachinePrecision] + 0.5), $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. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. frac-2neg N/A

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

   5. neg-mul-1 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. neg-mul-1 N/A

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

   10. frac-2neg N/A

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

   11. lift-/.f64 N/A

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

   12. flip3-+ N/A

       \[\leadsto \log \color{blue}{\left(\frac{{\left(\frac{1}{x}\right)}^{3} + {\left(\frac{\sqrt{1 - x \cdot x}}{x}\right)}^{3}}{\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{\sqrt{1 - x \cdot x}}{x} \cdot \frac{\sqrt{1 - x \cdot x}}{x} - \frac{1}{x} \cdot \frac{\sqrt{1 - x \cdot x}}{x}\right)}\right)} \]

4. Applied egg-rr 54.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}\right)\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{16} \cdot {x}^{2}\right) + \frac{1}{2}\right)}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{8} + \frac{1}{16} \cdot {x}^{2}, \frac{1}{2}\right)}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{8} + \frac{1}{16} \cdot {x}^{2}, \frac{1}{2}\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{8} + \frac{1}{16} \cdot {x}^{2}, \frac{1}{2}\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{16} \cdot {x}^{2} + \frac{1}{8}}, \frac{1}{2}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{16} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{8}, \frac{1}{2}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{16} \cdot x\right) \cdot x} + \frac{1}{8}, \frac{1}{2}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{16} \cdot x\right)} + \frac{1}{8}, \frac{1}{2}\right)\right)\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{16} \cdot x, \frac{1}{8}\right)}, \frac{1}{2}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{16}}, \frac{1}{8}\right), \frac{1}{2}\right)\right)\right) \]

   12. lower-*.f64 99.5

       \[\leadsto -\log \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0625}, 0.125\right), 0.5\right)\right) \]

7. Simplified 99.5%

   \[\leadsto -\log \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0625, 0.125\right), 0.5\right)\right)} \]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. distribute-lft-neg-in N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(x \cdot \frac{1}{2}\right), 2\right)}}{x}\right) \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \log \left(\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 2\right)}{x}\right) \]

   11. metadata-eval N/A

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

   12. lower-*.f64 99.5

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

5. Simplified 99.5%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ -\log \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.125, 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (log (* x (fma x (* x 0.125) 0.5)))))

C

double code(double x) {
	return -log((x * fma(x, (x * 0.125), 0.5)));
}

Julia

function code(x)
	return Float64(-log(Float64(x * fma(x, Float64(x * 0.125), 0.5))))
end

Wolfram

code[x_] := (-N[Log[N[(x * N[(x * N[(x * 0.125), $MachinePrecision] + 0.5), $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. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. frac-2neg N/A

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

   5. neg-mul-1 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. neg-mul-1 N/A

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

   10. frac-2neg N/A

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

   11. lift-/.f64 N/A

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

   12. flip3-+ N/A

       \[\leadsto \log \color{blue}{\left(\frac{{\left(\frac{1}{x}\right)}^{3} + {\left(\frac{\sqrt{1 - x \cdot x}}{x}\right)}^{3}}{\frac{1}{x} \cdot \frac{1}{x} + \left(\frac{\sqrt{1 - x \cdot x}}{x} \cdot \frac{\sqrt{1 - x \cdot x}}{x} - \frac{1}{x} \cdot \frac{\sqrt{1 - x \cdot x}}{x}\right)}\right)} \]

4. Applied egg-rr 54.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {x}^{2}\right)\right)}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {x}^{2}\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{1}{2}\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{8}} + \frac{1}{2}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{8} + \frac{1}{2}\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{8}\right)} + \frac{1}{2}\right)\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{8}, \frac{1}{2}\right)}\right)\right) \]

   7. lower-*.f64 99.5

      \[\leadsto -\log \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.125}, 0.5\right)\right) \]

7. Simplified 99.5%

   \[\leadsto -\log \color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.125, 0.5\right)\right)} \]
8. Add Preprocessing

Alternative 5: 99.1% accurate, 1.3× 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

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

   1. lower-/.f64 99.2

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

5. Simplified 99.2%

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

   1. clear-num N/A

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

   2. log-rec N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 99.2

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

7. Applied egg-rr 99.2%

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

Alternative 6: 0.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \log -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * N[Log[-1.0], $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 inf

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

   1. lower-sqrt.f64 0.0

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

5. Simplified 0.0%

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

   1. pow1/2 N/A

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

   2. pow-to-exp N/A

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

   3. rem-log-exp N/A

      \[\leadsto \color{blue}{\log -1 \cdot \frac{1}{2}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\log -1 \cdot \frac{1}{2}} \]

   5. lower-log.f64 0.0

      \[\leadsto \color{blue}{\log -1} \cdot 0.5 \]

7. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\log -1 \cdot 0.5} \]

8. Final simplification 0.0%

   \[\leadsto 0.5 \cdot \log -1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))