Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Log[N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $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. Add Preprocessing

3. Taylor expanded in x around 0 100.0%

   \[\leadsto \log \color{blue}{\left(\frac{2 + -0.5 \cdot {x}^{2}}{x}\right)} \]
4. Step-by-step derivation

   1. *-commutative 100.0%

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

5. Simplified 100.0%

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

7. Applied egg-rr 100.0%

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

Alternative 3: 31.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x - N[Log[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. Taylor expanded in x around -inf 0.0%

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

   1. +-commutative 0.0%

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

   2. mul-1-neg 0.0%

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

   3. unsub-neg 0.0%

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

5. Simplified 0.0%

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

7. Applied egg-rr 31.5%

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

8. Taylor expanded in x around 0 31.5%

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

   1. neg-mul-1 31.5%

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

   2. unsub-neg 31.5%

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

10. Simplified 31.5%

    \[\leadsto \color{blue}{x - \log x} \]
11. Add Preprocessing

Alternative 4: 31.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log1p (/ 1.0 x)))

C

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

Java

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

Python

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

Julia

function code(x)
	return log1p(Float64(1.0 / x))
end

Wolfram

code[x_] := N[Log[1 + N[(1.0 / 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. Taylor expanded in x around -inf 0.0%

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

   1. +-commutative 0.0%

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

   2. mul-1-neg 0.0%

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

   3. unsub-neg 0.0%

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

5. Simplified 0.0%

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

7. Applied egg-rr 31.5%

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

   1. *-un-lft-identity 31.5%

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

   2. log-prod 31.5%

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

   3. metadata-eval 31.5%

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

   4. sub-neg 31.5%

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

   5. metadata-eval 31.5%

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

   6. +-commutative 31.5%

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

   7. log1p-undefine 31.5%

      \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)} \]

9. Applied egg-rr 31.5%

   \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(\frac{1}{x}\right)} \]
10. Step-by-step derivation

    1. +-lft-identity 31.5%

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

11. Simplified 31.5%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{x}\right)} \]
12. Add Preprocessing

Alternative 5: 14.4% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ x + 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x 2.0))

C

double code(double x) {
	return x + 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return x + 2.0;
}

Python

def code(x):
	return x + 2.0

Julia

function code(x)
	return Float64(x + 2.0)
end

MATLAB

function tmp = code(x)
	tmp = x + 2.0;
end

Wolfram

code[x_] := N[(x + 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around -inf 0.0%

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

   1. +-commutative 0.0%

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

   2. mul-1-neg 0.0%

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

   3. unsub-neg 0.0%

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

5. Simplified 0.0%

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

7. Applied egg-rr 14.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{log1p}\left(x\right)\right)} - -1} \]
8. Step-by-step derivation

   1. sub-neg 14.4%

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

   2. metadata-eval 14.4%

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

   3. +-commutative 14.4%

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

   4. log1p-undefine 14.4%

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

   5. rem-exp-log 14.4%

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

   6. associate-+r+ 14.4%

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

   7. metadata-eval 14.4%

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

9. Simplified 14.4%

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

10. Taylor expanded in x around 0 14.4%

    \[\leadsto \color{blue}{2 + x} \]
11. Step-by-step derivation

    1. +-commutative 14.4%

       \[\leadsto \color{blue}{x + 2} \]

12. Simplified 14.4%

    \[\leadsto \color{blue}{x + 2} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))