Hyperbolic arc-cosine

Percentage Accurate: 52.0% → 100.0%

Time: 6.6s

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: 52.0% 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: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. pow1/2 53.8%

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

   2. difference-of-sqr-1 53.8%

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

   3. unpow-prod-down 100.0%

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

   4. sub-neg 100.0%

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

   5. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

   1. unpow1/2 100.0%

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

   2. unpow1/2 100.0%

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

6. Simplified 100.0%

   \[\leadsto \log \left(x + \color{blue}{\sqrt{x + 1} \cdot \sqrt{x + -1}}\right) \]
7. Add Preprocessing

Alternative 2: 99.1% 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 53.8%

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

3. Taylor expanded in x around inf 98.7%

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

Alternative 3: 31.4% 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 53.8%

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

3. Taylor expanded in x around inf 98.7%

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

4. Taylor expanded in x around 0 98.5%

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

6. Simplified 31.4%

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

Alternative 4: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

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

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 53.8%

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

   1. add-cbrt-cube 33.8%

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

   2. pow1/3 33.6%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 33.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   4. pow3 33.6%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(x + \sqrt{x \cdot x - 1}\right)}^{3}\right)} \]

   5. log-pow 53.6%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   6. fma-neg 53.6%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right) \]

   7. metadata-eval 53.6%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right) \]

4. Applied egg-rr 53.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

5. Taylor expanded in x around 0 0.0%

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

7. Simplified 1.1%

   \[\leadsto \color{blue}{-2 + \frac{x}{-2}} \]

8. Taylor expanded in x around 0 1.6%

   \[\leadsto \color{blue}{-2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))