Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 5

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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-1 + \frac{1 + \sqrt{1 - {x}^{2}}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log1p (+ -1.0 (/ (+ 1.0 (sqrt (- 1.0 (pow x 2.0)))) x))))

C

double code(double x) {
	return log1p((-1.0 + ((1.0 + sqrt((1.0 - pow(x, 2.0)))) / x)));
}

Java

public static double code(double x) {
	return Math.log1p((-1.0 + ((1.0 + Math.sqrt((1.0 - Math.pow(x, 2.0)))) / x)));
}

Python

def code(x):
	return math.log1p((-1.0 + ((1.0 + math.sqrt((1.0 - math.pow(x, 2.0)))) / x)))

Julia

function code(x)
	return log1p(Float64(-1.0 + Float64(Float64(1.0 + sqrt(Float64(1.0 - (x ^ 2.0)))) / x)))
end

Wolfram

code[x_] := N[Log[1 + N[(-1.0 + N[(N[(1.0 + N[Sqrt[N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $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. Step-by-step derivation

   1. log1p-expm1-u 100.0%

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

   2. expm1-undefine 100.0%

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

   3. add-exp-log 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. div-inv 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. fmm-def 100.0%

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

   8. pow2 100.0%

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

   9. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{x}, 1 + \sqrt{1 - {x}^{2}}, -1\right)\right)} \]
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-lft-identity 100.0%

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

6. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 + \frac{1 + \sqrt{1 - {x}^{2}}}{x}\right)} \]
7. 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

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{2 + x \cdot \left(-1 + x \cdot -0.5\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log1p (/ (+ 2.0 (* x (+ -1.0 (* x -0.5)))) x)))

C

double code(double x) {
	return log1p(((2.0 + (x * (-1.0 + (x * -0.5)))) / x));
}

Java

public static double code(double x) {
	return Math.log1p(((2.0 + (x * (-1.0 + (x * -0.5)))) / x));
}

Python

def code(x):
	return math.log1p(((2.0 + (x * (-1.0 + (x * -0.5)))) / x))

Julia

function code(x)
	return log1p(Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * -0.5)))) / x))
end

Wolfram

code[x_] := N[Log[1 + N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * -0.5), $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. log1p-expm1-u 100.0%

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

   2. expm1-undefine 100.0%

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

   3. add-exp-log 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. div-inv 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. fmm-def 100.0%

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

   8. pow2 100.0%

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

   9. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{x}, 1 + \sqrt{1 - {x}^{2}}, -1\right)\right)} \]
5. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-lft-identity 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in x around 0 99.3%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{2 + x \cdot \left(-0.5 \cdot x - 1\right)}{x}}\right) \]

8. Final simplification 99.3%

   \[\leadsto \mathsf{log1p}\left(\frac{2 + x \cdot \left(-1 + x \cdot -0.5\right)}{x}\right) \]
9. Add Preprocessing

Alternative 4: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 99.3%

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

   1. *-commutative 99.3%

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

5. Simplified 99.3%

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

   1. unpow2 99.3%

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

7. Applied egg-rr 99.3%

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

8. Taylor expanded in x around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. fma-undefine 99.3%

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

   4. *-lft-identity 99.3%

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

   5. associate-*l/ 99.3%

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

   6. fma-undefine 99.3%

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

   7. +-commutative 99.3%

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

   8. distribute-rgt-in 99.3%

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

   9. associate-*r/ 99.3%

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

   10. metadata-eval 99.3%

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

   11. associate-*r/ 99.3%

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

   12. *-rgt-identity 99.3%

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

   13. *-commutative 99.3%

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

   14. associate-*r/ 99.3%

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

   15. *-lft-identity 99.3%

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

   16. associate-*l/ 99.3%

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

   17. unpow2 99.3%

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

   18. associate-*r* 99.3%

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

   19. lft-mult-inverse 99.3%

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

   20. *-lft-identity 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

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. Add Preprocessing

3. Taylor expanded in x around 0 98.8%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))