Hyperbolic arc-cosine

Percentage Accurate: 51.4% → 98.8%

Time: 5.1s

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: 51.4% 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: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in x around inf 98.6%

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

   1. count-2 98.6%

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

   2. add-sqr-sqrt 98.6%

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

   3. associate-*r* 98.6%

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

   4. log-prod 98.8%

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

5. Applied egg-rr 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. log-prod 98.8%

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

   3. pow1/2 98.8%

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

   4. sqrt-pow1 98.8%

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

   5. log-pow 98.8%

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

   6. metadata-eval 98.8%

      \[\leadsto \log \left(2 \cdot \sqrt{x}\right) + \left(\color{blue}{0.25} \cdot \log x + \log \left(\sqrt{\sqrt{x}}\right)\right) \]

   7. pow1/2 98.8%

      \[\leadsto \log \left(2 \cdot \sqrt{x}\right) + \left(0.25 \cdot \log x + \log \left(\sqrt{\color{blue}{{x}^{0.5}}}\right)\right) \]

   8. sqrt-pow1 98.8%

      \[\leadsto \log \left(2 \cdot \sqrt{x}\right) + \left(0.25 \cdot \log x + \log \color{blue}{\left({x}^{\left(\frac{0.5}{2}\right)}\right)}\right) \]

   9. log-pow 98.8%

      \[\leadsto \log \left(2 \cdot \sqrt{x}\right) + \left(0.25 \cdot \log x + \color{blue}{\frac{0.5}{2} \cdot \log x}\right) \]

   10. metadata-eval 98.8%

       \[\leadsto \log \left(2 \cdot \sqrt{x}\right) + \left(0.25 \cdot \log x + \color{blue}{0.25} \cdot \log x\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto \log \left(2 \cdot \sqrt{x}\right) + \color{blue}{\left(0.25 \cdot \log x + 0.25 \cdot \log x\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-out 98.8%

      \[\leadsto \log \left(2 \cdot \sqrt{x}\right) + \color{blue}{\log x \cdot \left(0.25 + 0.25\right)} \]

   2. metadata-eval 98.8%

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

9. Simplified 98.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.0%

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

3. Taylor expanded in x around inf 98.7%

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

   1. mul-1-neg 98.7%

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

   2. log-rec 98.7%

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

   3. remove-double-neg 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

   \[\leadsto \log x + \log 2 \]
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 52.0%

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

3. Taylor expanded in x around inf 98.6%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))