Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

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: 100.0% 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: 100.0% 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 2: 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. Add Preprocessing

3. Taylor expanded in x around 0 99.7%

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

   1. *-commutative 99.7%

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

5. Simplified 99.7%

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

   1. unpow2 99.7%

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

7. Applied egg-rr 99.7%

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

Alternative 3: 99.1% 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 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 99.1%

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

   1. clear-num 99.1%

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

   2. log-rec 99.1%

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

   3. div-inv 99.1%

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

   4. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

Alternative 4: 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 99.1%

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

Alternative 5: 3.1% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

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 99.1%

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

   1. clear-num 99.1%

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

   2. log-rec 99.1%

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

   3. div-inv 99.1%

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

   4. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

   1. add-exp-log 99.1%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 1.5%

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

   4. sqr-neg 1.5%

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

   5. sqrt-unprod 1.5%

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

   6. add-sqr-sqrt 1.5%

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

   7. neg-log 1.5%

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

   8. *-commutative 1.5%

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

   9. associate-/r* 1.5%

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

   10. metadata-eval 1.5%

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

   11. add-exp-log 1.5%

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

   12. clear-num 1.5%

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

   13. div-inv 1.5%

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

   14. metadata-eval 1.5%

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

   15. add-sqr-sqrt 1.5%

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

   16. associate-/r* 1.5%

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

7. Applied egg-rr 3.1%

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

   1. *-inverses 3.1%

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

9. Simplified 3.1%

   \[\leadsto -\log \color{blue}{1} \]
10. Step-by-step derivation

    1. metadata-eval 3.1%

       \[\leadsto -\color{blue}{0} \]

    2. metadata-eval 3.1%

       \[\leadsto \color{blue}{0} \]

11. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024182 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))