Hyperbolic arc-cosine

Percentage Accurate: 52.6% → 98.6%

Time: 5.2s

Alternatives: 5

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: 52.6% 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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

3. Taylor expanded in x around inf 99.3%

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

   1. mul-1-neg 99.3%

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

   2. log-rec 99.3%

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

   3. remove-double-neg 99.3%

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

5. Simplified 99.3%

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

Alternative 2: 98.9% 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 49.3%

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

3. Taylor expanded in x around inf 99.2%

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

Alternative 3: 31.3% accurate, 2.0× 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 log(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.3%

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

3. Taylor expanded in x around inf 99.2%

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

4. Taylor expanded in x around 0 99.3%

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

6. Simplified 31.6%

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

Alternative 4: 14.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log 4 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log 4.0))

C

double code(double x) {
	return log(4.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(4.0d0)
end function

Java

public static double code(double x) {
	return Math.log(4.0);
}

Python

def code(x):
	return math.log(4.0)

Julia

function code(x)
	return log(4.0)
end

MATLAB

function tmp = code(x)
	tmp = log(4.0);
end

Wolfram

code[x_] := N[Log[4.0], $MachinePrecision]

Derivation

1. Initial program 49.3%

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

3. Taylor expanded in x around inf 99.2%

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

   1. flip-+ 0.0%

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

   2. difference-of-squares 0.0%

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

   3. +-inverses 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. associate-*r/ 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

   10. flip-+ 10.6%

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

   11. pow2 10.6%

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

5. Applied egg-rr 0.0%

   \[\leadsto \log \color{blue}{\left(4 \cdot {\left(x \cdot \frac{0}{0}\right)}^{2}\right)} \]

7. Simplified 14.2%

   \[\leadsto \log \color{blue}{4} \]
8. Add Preprocessing

Alternative 5: 13.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 49.3%

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

3. Taylor expanded in x around inf 99.2%

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

   1. flip-+ 0.0%

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

   2. +-inverses 0.0%

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

   3. +-inverses 0.0%

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

   4. diff-log 0.0%

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

   5. +-inverses 0.0%

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

   6. metadata-eval 0.0%

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

   7. +-inverses 0.0%

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

   8. pow1/2 0.0%

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

   9. +-inverses 0.0%

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

   10. metadata-eval 0.0%

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

   11. +-inverses 0.0%

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

   12. pow1/2 0.0%

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

   13. log-div 0.0%

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

   14. +-inverses 0.0%

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

   15. +-inverses 0.0%

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

   16. sqrt-div 0.0%

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

   17. flip-+ 18.7%

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

   18. pow1/2 18.7%

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

   19. log-pow 18.7%

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

5. Applied egg-rr 0.0%

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

7. Simplified 13.8%

   \[\leadsto \color{blue}{0.5} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))