Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 4.9s

Alternatives: 6

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: 99.1% 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. Add Preprocessing

3. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 14.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{-1 + \log 2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(-1.0 / N[(-1.0 + N[Log[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. Add Preprocessing

3. Taylor expanded in x around 0 99.5%

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

   1. count-2 99.5%

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

   2. log-prod 99.2%

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

   3. neg-log 99.2%

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

   4. flip-+ 98.9%

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

   5. pow2 98.9%

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

5. Applied egg-rr 98.9%

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

7. Simplified 14.6%

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

8. Final simplification 14.6%

   \[\leadsto \frac{-1}{-1 + \log 2} \]
9. Add Preprocessing

Alternative 4: 5.4% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (1.0 / x) + -1.0

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] + -1.0), $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. *-un-lft-identity 100.0%

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

   2. log-prod 100.0%

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

   3. metadata-eval 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. div-inv 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. pow2 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - {x}^{2}}\right)\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. log-div 99.7%

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

   4. *-lft-identity 99.7%

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

   5. log1p-def 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around inf 0.0%

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

9. Simplified 5.3%

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

10. Taylor expanded in x around 0 5.3%

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

11. Final simplification 5.3%

    \[\leadsto \frac{1}{x} + -1 \]
12. Add Preprocessing

Alternative 5: 5.4% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / 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. *-un-lft-identity 100.0%

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

   2. log-prod 100.0%

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

   3. metadata-eval 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. div-inv 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. pow2 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - {x}^{2}}\right)\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. log-div 99.7%

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

   4. *-lft-identity 99.7%

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

   5. log1p-def 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around inf 0.0%

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

9. Simplified 5.3%

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

10. Taylor expanded in x around 0 5.3%

    \[\leadsto \color{blue}{\frac{1}{x}} \]

11. Final simplification 5.3%

    \[\leadsto \frac{1}{x} \]
12. Add Preprocessing

Alternative 6: 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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 100.0%

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

   2. log-prod 100.0%

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

   3. metadata-eval 100.0%

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

   4. *-un-lft-identity 100.0%

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

   5. div-inv 100.0%

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

   6. distribute-rgt-out 100.0%

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

   7. pow2 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \log \left(\frac{1}{x} \cdot \left(1 + \sqrt{1 - {x}^{2}}\right)\right)} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*l/ 100.0%

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

   3. log-div 99.7%

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

   4. *-lft-identity 99.7%

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

   5. log1p-def 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around inf 0.0%

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

9. Simplified 1.6%

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

10. Final simplification 1.6%

    \[\leadsto -1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))