Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 4.9s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Log[N[(N[(1.0 + N[Sqrt[N[(x * (-x) + 1.0), $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 100.0%

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

   2. expm1-udef 100.0%

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

   3. +-commutative 100.0%

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

   4. div-inv 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. cancel-sign-sub-inv 100.0%

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

   8. +-commutative 100.0%

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

   9. *-commutative 100.0%

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

   10. fma-def 100.0%

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

3. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-lft-identity 100.0%

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

   5. +-commutative 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

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. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(N[(x * -0.5), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * 2.0), $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 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 + N[Log[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $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 99.1%

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

   1. log1p-expm1-u 99.1%

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

   2. expm1-udef 99.1%

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

   3. add-exp-log 99.1%

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

4. Applied egg-rr 99.1%

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

   1. expm1-log1p-u 97.4%

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

   2. expm1-udef 97.4%

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

   3. log1p-udef 97.4%

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

   4. add-exp-log 99.1%

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

   5. add-exp-log 99.1%

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

   6. expm1-def 99.1%

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

   7. log1p-expm1-u 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 5: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := N[Log[1 + N[(N[(2.0 / x), $MachinePrecision] + -1.0), $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 99.1%

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

   1. log1p-expm1-u 99.1%

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

   2. expm1-udef 99.1%

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

   3. add-exp-log 99.1%

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

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 6: 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. Taylor expanded in x around 0 99.1%

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

   1. clear-num 99.1%

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

   2. log-div 99.1%

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

   3. metadata-eval 99.1%

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

   4. div-inv 99.1%

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

   5. metadata-eval 99.1%

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

4. Applied egg-rr 99.1%

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

   1. neg-sub0 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 7: 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. Taylor expanded in x around 0 99.1%

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

3. Final simplification 99.1%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))