Hyperbolic arc-cosine

Percentage Accurate: 51.8% → 99.5%

Time: 6.1s

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: 51.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in x around inf 98.6%

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

   1. *-commutative 98.6%

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

   2. associate-*r/ 98.6%

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

   3. metadata-eval 98.6%

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 2: 31.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. add-cbrt-cube 31.5%

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

   2. pow3 31.5%

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

   3. sqrt-pow2 31.5%

      \[\leadsto \log \left(x + \sqrt[3]{\color{blue}{{\left(x \cdot x - 1\right)}^{\left(\frac{3}{2}\right)}}}\right) \]

   4. fma-neg 31.5%

      \[\leadsto \log \left(x + \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{\left(\frac{3}{2}\right)}}\right) \]

   5. metadata-eval 31.5%

      \[\leadsto \log \left(x + \sqrt[3]{{\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{\left(\frac{3}{2}\right)}}\right) \]

   6. metadata-eval 31.5%

      \[\leadsto \log \left(x + \sqrt[3]{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{\color{blue}{1.5}}}\right) \]

4. Applied egg-rr 31.5%

   \[\leadsto \log \left(x + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{1.5}}}\right) \]

5. Taylor expanded in x around 0 0.0%

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

7. Simplified 31.5%

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

8. Final simplification 31.5%

   \[\leadsto \log \left(x + -1\right) \]
9. Add Preprocessing

Alternative 3: 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 50.0%

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

3. Taylor expanded in x around inf 98.2%

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

4. Final simplification 98.2%

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

Alternative 4: 1.5% 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 Float64(-log(x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

3. Taylor expanded in x around inf 98.2%

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

4. Taylor expanded in x around inf 98.2%

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

6. Simplified 1.5%

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

7. Final simplification 1.5%

   \[\leadsto -\log x \]
8. Add Preprocessing

Alternative 5: 1.1% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ -1 + \frac{x}{-1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ -1.0 (/ x -1.0)))

C

double code(double x) {
	return -1.0 + (x / -1.0);
}

Fortran

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

Java

public static double code(double x) {
	return -1.0 + (x / -1.0);
}

Python

def code(x):
	return -1.0 + (x / -1.0)

Julia

function code(x)
	return Float64(-1.0 + Float64(x / -1.0))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 + (x / -1.0);
end

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. add-cbrt-cube 31.5%

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

   2. pow3 31.5%

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

   3. sqrt-pow2 31.5%

      \[\leadsto \log \left(x + \sqrt[3]{\color{blue}{{\left(x \cdot x - 1\right)}^{\left(\frac{3}{2}\right)}}}\right) \]

   4. fma-neg 31.5%

      \[\leadsto \log \left(x + \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{\left(\frac{3}{2}\right)}}\right) \]

   5. metadata-eval 31.5%

      \[\leadsto \log \left(x + \sqrt[3]{{\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{\left(\frac{3}{2}\right)}}\right) \]

   6. metadata-eval 31.5%

      \[\leadsto \log \left(x + \sqrt[3]{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{\color{blue}{1.5}}}\right) \]

4. Applied egg-rr 31.5%

   \[\leadsto \log \left(x + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{1.5}}}\right) \]

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%

   \[\leadsto \color{blue}{-1 + \frac{x}{-1}} \]

8. Final simplification 1.1%

   \[\leadsto -1 + \frac{x}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024045 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))