Hyperbolic arc-cosine

Percentage Accurate: 51.5% → 99.5%

Time: 6.9s

Alternatives: 6

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around inf 99.6%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 - 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r/ 99.6%

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

   2. metadata-eval 99.6%

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

5. Simplified 99.6%

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

Alternative 2: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
4. Add Preprocessing

Alternative 3: 14.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log 2.125 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log 2.125))

C

double code(double x) {
	return log(2.125);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(2.125d0)
end function

Java

public static double code(double x) {
	return Math.log(2.125);
}

Python

def code(x):
	return math.log(2.125)

Julia

function code(x)
	return log(2.125)
end

MATLAB

function tmp = code(x)
	tmp = log(2.125);
end

Wolfram

code[x_] := N[Log[2.125], $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow2 51.2%

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

   2. add-sqr-sqrt 51.2%

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

   3. pow2 51.2%

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

   4. pow-pow 51.2%

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

   5. metadata-eval 51.2%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 51.2%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around -inf 0.0%

   \[\leadsto \log \color{blue}{\left(-1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{0.5 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + 0.125 \cdot \frac{1}{{x}^{2} \cdot {\left(\sqrt{-1}\right)}^{6}}}{{x}^{2}} + {\left(\sqrt{-1}\right)}^{2}\right) - 1\right)\right)\right)} \]

7. Simplified 14.0%

   \[\leadsto \log \color{blue}{2.125} \]
8. Add Preprocessing

Alternative 4: 5.4% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ -8 + x \cdot 0.8333333333333334 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -8.0 (* x 0.8333333333333334)))

C

double code(double x) {
	return -8.0 + (x * 0.8333333333333334);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-8.0d0) + (x * 0.8333333333333334d0)
end function

Java

public static double code(double x) {
	return -8.0 + (x * 0.8333333333333334);
}

Python

def code(x):
	return -8.0 + (x * 0.8333333333333334)

Julia

function code(x)
	return Float64(-8.0 + Float64(x * 0.8333333333333334))
end

MATLAB

function tmp = code(x)
	tmp = -8.0 + (x * 0.8333333333333334);
end

Wolfram

code[x_] := N[(-8.0 + N[(x * 0.8333333333333334), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow2 51.2%

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

   2. add-sqr-sqrt 51.2%

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

   3. pow2 51.2%

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

   4. pow-pow 51.2%

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

   5. metadata-eval 51.2%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 51.2%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around 0 0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left(-0.16666666666666666 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{3}} + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 5.6%

   \[\leadsto \color{blue}{-8 + x \cdot 0.8333333333333334} \]
8. Add Preprocessing

Alternative 5: 5.4% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.8333333333333334 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.8333333333333334))

C

double code(double x) {
	return x * 0.8333333333333334;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.8333333333333334d0
end function

Java

public static double code(double x) {
	return x * 0.8333333333333334;
}

Python

def code(x):
	return x * 0.8333333333333334

Julia

function code(x)
	return Float64(x * 0.8333333333333334)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.8333333333333334;
end

Wolfram

code[x_] := N[(x * 0.8333333333333334), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow2 51.2%

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

   2. add-sqr-sqrt 51.2%

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

   3. pow2 51.2%

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

   4. pow-pow 51.2%

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

   5. metadata-eval 51.2%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 51.2%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around 0 0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left(-0.16666666666666666 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{3}} + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 5.6%

   \[\leadsto \color{blue}{-8 + x \cdot 0.8333333333333334} \]

8. Taylor expanded in x around inf 5.6%

   \[\leadsto \color{blue}{x \cdot \left(0.8333333333333334 - 8 \cdot \frac{1}{x}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 5.6%

      \[\leadsto x \cdot \left(0.8333333333333334 - \color{blue}{\frac{8 \cdot 1}{x}}\right) \]

   2. metadata-eval 5.6%

      \[\leadsto x \cdot \left(0.8333333333333334 - \frac{\color{blue}{8}}{x}\right) \]

10. Simplified 5.6%

    \[\leadsto \color{blue}{x \cdot \left(0.8333333333333334 - \frac{8}{x}\right)} \]

11. Taylor expanded in x around inf 5.6%

    \[\leadsto x \cdot \color{blue}{0.8333333333333334} \]
12. Add Preprocessing

Alternative 6: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -8.0)

C

double code(double x) {
	return -8.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -8.0d0
end function

Java

public static double code(double x) {
	return -8.0;
}

Python

def code(x):
	return -8.0

Julia

function code(x)
	return -8.0
end

MATLAB

function tmp = code(x)
	tmp = -8.0;
end

Wolfram

code[x_] := -8.0

Derivation

1. Initial program 51.2%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow2 51.2%

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

   2. add-sqr-sqrt 51.2%

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

   3. pow2 51.2%

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

   4. pow-pow 51.2%

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

   5. metadata-eval 51.2%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt{x}\right)}^{\color{blue}{4}} - 1}\right) \]

4. Applied egg-rr 51.2%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt{x}\right)}^{4}} - 1}\right) \]

5. Taylor expanded in x around 0 0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left(-0.16666666666666666 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{3}} + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 5.6%

   \[\leadsto \color{blue}{-8 + x \cdot 0.8333333333333334} \]

8. Taylor expanded in x around 0 1.6%

   \[\leadsto \color{blue}{-8} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))