Hyperbolic arc-cosine

Percentage Accurate: 51.7% → 98.7%

Time: 5.9s

Alternatives: 4

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: 51.7% 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: 98.7% 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 50.1%

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

3. Taylor expanded in x around inf 99.0%

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

   1. mul-1-neg 99.0%

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

   2. log-rec 99.0%

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

   3. remove-double-neg 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

   \[\leadsto \log 2 + \log x \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in x around inf 98.9%

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

   1. *-commutative 98.9%

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

   2. associate-*r/ 98.9%

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

   3. metadata-eval 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 3: 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 50.1%

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

3. Taylor expanded in x around inf 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 4: 1.6% accurate, 207.0× 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 50.1%

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

3. Taylor expanded in x around inf 98.5%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

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

   7. +-inverses 0.0%

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

   8. flip-+ 10.7%

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

   9. count-2 10.7%

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

   10. *-commutative 10.7%

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

   11. sum-log 18.6%

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

   12. *-commutative 18.6%

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

   13. sum-log 18.6%

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

   14. *-commutative 18.6%

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

   15. sum-log 18.7%

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

   16. sum-log 18.6%

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

   17. count-2 18.6%

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

   18. flip-+ 0.0%

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

   19. +-inverses 0.0%

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

   20. +-inverses 0.0%

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

   21. sum-log 0.0%

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

   22. count-2 0.0%

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

5. Applied egg-rr 0.0%

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

7. Simplified 1.6%

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

8. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2024016 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))