Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 2.7s

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. Final simplification 100.0%

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

Alternative 2: 99.6% 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. Taylor expanded in x around 0 99.7%

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

3. Final simplification 99.7%

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

Alternative 3: 99.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ (/ 1.0 x) (/ 1.0 x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 99.5%

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

3. Final simplification 99.5%

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

Alternative 4: 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. Taylor expanded in x around 0 99.5%

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

   1. add-sqr-sqrt 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. distribute-lft-out 99.5%

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

   4. log-prod 99.2%

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

   5. inv-pow 99.2%

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

   6. sqrt-pow1 99.1%

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

   7. metadata-eval 99.1%

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

   8. log-pow 99.1%

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

   9. inv-pow 99.1%

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

   10. sqrt-pow1 99.1%

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

   11. metadata-eval 99.1%

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

   12. inv-pow 99.1%

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

   13. sqrt-pow1 99.1%

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

   14. metadata-eval 99.1%

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

4. Applied egg-rr 99.1%

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

6. Simplified 31.4%

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

7. Final simplification 31.4%

   \[\leadsto -\log x \]

Alternative 5: 1.6% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

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 99.7%

   \[\leadsto \log \left(\frac{1}{x} + \color{blue}{\left(-0.5 \cdot x + \frac{1}{x}\right)}\right) \]
3. Step-by-step derivation

   1. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. log-prod 99.7%

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

   4. +-commutative 99.7%

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

   5. associate-+l+ 99.7%

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

   6. *-commutative 99.7%

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

   7. fma-def 99.7%

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

   8. flip-+ 0.0%

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

   9. +-inverses 0.0%

      \[\leadsto \log \left(\mathsf{fma}\left(x, -0.5, \frac{\color{blue}{0}}{\frac{1}{x} - \frac{1}{x}}\right)\right) + \log 1 \]

   10. +-inverses 0.0%

       \[\leadsto \log \left(\mathsf{fma}\left(x, -0.5, \frac{0}{\color{blue}{0}}\right)\right) + \log 1 \]

   11. metadata-eval 0.0%

       \[\leadsto \log \left(\mathsf{fma}\left(x, -0.5, \frac{0}{0}\right)\right) + \color{blue}{0} \]

4. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(x, -0.5, \frac{0}{0}\right)\right) + 0} \]

6. Simplified 1.6%

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

7. Final simplification 1.6%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))