Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 5.4s

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, 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. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(N[(1.0 / x), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $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 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \log \left(\frac{1}{x} + \left(\frac{1}{x} + x \cdot -0.5\right)\right) \]
5. Add Preprocessing

Alternative 3: 99.0% 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. Step-by-step derivation

   1. expm1-log1p-u 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. div-inv 98.2%

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

   4. distribute-rgt-out 98.2%

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

   5. pow2 98.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - \color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

4. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - {x}^{2}}\right)\right)\right)\right)} \]

5. Taylor expanded in x around 0 97.4%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \color{blue}{\left(\frac{2}{x}\right)}\right)\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.1%

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

   2. clear-num 99.1%

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

   3. log-div 99.1%

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

   4. metadata-eval 99.1%

      \[\leadsto \color{blue}{0} - \log \left(\frac{x}{2}\right) \]

   5. div-inv 99.1%

      \[\leadsto 0 - \log \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

   6. metadata-eval 99.1%

      \[\leadsto 0 - \log \left(x \cdot \color{blue}{0.5}\right) \]

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{0 - \log \left(x \cdot 0.5\right)} \]
8. Step-by-step derivation

   1. neg-sub0 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

    \[\leadsto -\log \left(x \cdot 0.5\right) \]
11. Add Preprocessing

Alternative 4: 99.0% 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. Step-by-step derivation

   1. expm1-log1p-u 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. div-inv 98.2%

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

   4. distribute-rgt-out 98.2%

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

   5. pow2 98.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - \color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

4. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - {x}^{2}}\right)\right)\right)\right)} \]

5. Taylor expanded in x around 0 97.4%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \color{blue}{\left(\frac{2}{x}\right)}\right)\right) \]

6. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{\log 2 + -1 \cdot \log x} \]
7. Step-by-step derivation

   1. neg-mul-1 98.8%

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

   2. sub-neg 98.8%

      \[\leadsto \color{blue}{\log 2 - \log x} \]

   3. log-div 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 5: 31.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ -\log x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (log x)))

C

double code(double x) {
	return -log(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log(x)
end function

Java

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

Python

def code(x):
	return -math.log(x)

Julia

function code(x)
	return Float64(-log(x))
end

MATLAB

function tmp = code(x)
	tmp = -log(x);
end

Wolfram

code[x_] := (-N[Log[x], $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. expm1-log1p-u 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. div-inv 98.2%

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

   4. distribute-rgt-out 98.2%

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

   5. pow2 98.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - \color{blue}{{x}^{2}}}\right)\right)\right)\right) \]

4. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - {x}^{2}}\right)\right)\right)\right)} \]

5. Taylor expanded in x around 0 98.8%

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

7. Simplified 31.4%

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

8. Final simplification 31.4%

   \[\leadsto -\log x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))