Hyperbolic arc-cosine

Percentage Accurate: 51.2% → 99.5%

Time: 4.5s

Alternatives: 3

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: 51.2% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

2. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

4. Simplified 99.4%

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

5. Final simplification 99.4%

   \[\leadsto \log \left(x + \left(x - \frac{0.5}{x}\right)\right) \]

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 50.0%

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

2. Taylor expanded in x around inf 99.0%

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

3. Final simplification 99.0%

   \[\leadsto \log \left(x + x\right) \]

Alternative 3: 0.0% 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 50.0%

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

2. Taylor expanded in x around inf 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

4. Simplified 99.4%

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

   1. associate-+r- 99.4%

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

   2. flip-- 49.5%

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

   3. frac-times 49.5%

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

   4. metadata-eval 49.5%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x + x\right) - \frac{\color{blue}{0.25}}{x \cdot x}}{\left(x + x\right) + \frac{0.5}{x}}\right) \]

6. Applied egg-rr 49.5%

   \[\leadsto \log \color{blue}{\left(\frac{\left(x + x\right) \cdot \left(x + x\right) - \frac{0.25}{x \cdot x}}{\left(x + x\right) + \frac{0.5}{x}}\right)} \]
7. Step-by-step derivation

   1. metadata-eval 49.5%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x + x\right) - \frac{0.25}{x \cdot x}}{\left(x + x\right) + \frac{\color{blue}{0.5 \cdot 1}}{x}}\right) \]

   2. associate-*r/ 49.5%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x + x\right) - \frac{0.25}{x \cdot x}}{\left(x + x\right) + \color{blue}{0.5 \cdot \frac{1}{x}}}\right) \]

   3. +-commutative 49.5%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x + x\right) - \frac{0.25}{x \cdot x}}{\color{blue}{0.5 \cdot \frac{1}{x} + \left(x + x\right)}}\right) \]

   4. associate-*r/ 49.5%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x + x\right) - \frac{0.25}{x \cdot x}}{\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(x + x\right)}\right) \]

   5. metadata-eval 49.5%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x + x\right) - \frac{0.25}{x \cdot x}}{\frac{\color{blue}{0.5}}{x} + \left(x + x\right)}\right) \]

8. Simplified 49.5%

   \[\leadsto \log \color{blue}{\left(\frac{\left(x + x\right) \cdot \left(x + x\right) - \frac{0.25}{x \cdot x}}{\frac{0.5}{x} + \left(x + x\right)}\right)} \]

9. Taylor expanded in x around 0 0.0%

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

11. Simplified 0.0%

    \[\leadsto \log \color{blue}{-4} \]

12. Final simplification 0.0%

    \[\leadsto \log -4 \]

Reproduce

herbie shell --seed 2023181 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))