Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 100.0%

Time: 7.1s

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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := (-N[Log[N[(x / N[(1.0 + N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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. expm1-log1p-u 99.9%

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

   2. expm1-udef 99.9%

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

   3. +-commutative 99.9%

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

   4. div-inv 99.9%

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

   5. *-un-lft-identity 99.9%

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

   6. distribute-rgt-out 99.9%

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

3. Applied egg-rr 99.9%

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

   1. expm1-def 99.9%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-lft-identity 100.0%

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

   5. +-commutative 100.0%

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

5. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. log-rec 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× 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. Step-by-step derivation

   1. expm1-log1p-u 99.9%

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

   2. expm1-udef 99.9%

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

   3. +-commutative 99.9%

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

   4. div-inv 99.9%

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

   5. *-un-lft-identity 99.9%

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

   6. distribute-rgt-out 99.9%

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

3. Applied egg-rr 99.9%

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

   1. expm1-def 99.9%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-lft-identity 100.0%

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

   5. +-commutative 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((x * (-0.5d0)) + (2.0d0 * (1.0d0 / x))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = log(((x * -0.5) + (2.0 * (1.0 / x))));
end

Wolfram

code[x_] := N[Log[N[(N[(x * -0.5), $MachinePrecision] + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 99.2%

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

3. Final simplification 99.2%

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

Alternative 4: 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. Taylor expanded in x around 0 98.6%

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

   1. clear-num 98.6%

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

   2. log-div 98.6%

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

   3. metadata-eval 98.6%

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

   4. div-inv 98.6%

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

   5. metadata-eval 98.6%

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

4. Applied egg-rr 98.6%

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

   1. neg-sub0 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

   \[\leadsto -\log \left(x \cdot 0.5\right) \]

Alternative 5: 98.9% 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. Taylor expanded in x around 0 98.6%

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

3. Final simplification 98.6%

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

Reproduce

herbie shell --seed 2023216 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))