Hyperbolic arc-cosine

Percentage Accurate: 52.5% → 99.4%

Time: 5.3s

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: 52.5% 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: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot 2 + \left(-1 - \frac{0.5}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (+ (* x 2.0) (- -1.0 (/ 0.5 x)))))

C

double code(double x) {
	return log1p(((x * 2.0) + (-1.0 - (0.5 / x))));
}

Java

public static double code(double x) {
	return Math.log1p(((x * 2.0) + (-1.0 - (0.5 / x))));
}

Python

def code(x):
	return math.log1p(((x * 2.0) + (-1.0 - (0.5 / x))))

Julia

function code(x)
	return log1p(Float64(Float64(x * 2.0) + Float64(-1.0 - Float64(0.5 / x))))
end

Wolfram

code[x_] := N[Log[1 + N[(N[(x * 2.0), $MachinePrecision] + N[(-1.0 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.7%

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

   1. log1p-expm1-u 52.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

   2. expm1-udef 52.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(x + \sqrt{x \cdot x - 1}\right)} - 1}\right) \]

   3. add-exp-log 52.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x + \sqrt{x \cdot x - 1}\right)} - 1\right) \]

   4. fma-neg 52.7%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) - 1\right) \]

   5. metadata-eval 52.7%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) - 1\right) \]

3. Applied egg-rr 52.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]

4. Taylor expanded in x around inf 99.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 1\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 99.2%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot 2} - \left(0.5 \cdot \frac{1}{x} + 1\right)\right) \]

   2. +-commutative 99.2%

      \[\leadsto \mathsf{log1p}\left(x \cdot 2 - \color{blue}{\left(1 + 0.5 \cdot \frac{1}{x}\right)}\right) \]

   3. associate-*r/ 99.2%

      \[\leadsto \mathsf{log1p}\left(x \cdot 2 - \left(1 + \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]

   4. metadata-eval 99.2%

      \[\leadsto \mathsf{log1p}\left(x \cdot 2 - \left(1 + \frac{\color{blue}{0.5}}{x}\right)\right) \]

6. Simplified 99.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot 2 - \left(1 + \frac{0.5}{x}\right)}\right) \]

7. Final simplification 99.2%

   \[\leadsto \mathsf{log1p}\left(x \cdot 2 + \left(-1 - \frac{0.5}{x}\right)\right) \]

Alternative 2: 99.4% 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 52.7%

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

2. Taylor expanded in x around inf 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.2%

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

   3. metadata-eval 99.2%

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

4. Simplified 99.2%

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

5. Final simplification 99.2%

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

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 52.7%

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

2. Taylor expanded in x around inf 99.1%

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

3. Final simplification 99.1%

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

Alternative 4: 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 52.7%

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

   1. add-cube-cbrt 52.7%

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

   2. pow3 52.7%

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

   3. pow1/3 52.6%

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

   4. pow1/2 52.6%

      \[\leadsto \log \left(x + {\left({\color{blue}{\left({\left(x \cdot x - 1\right)}^{0.5}\right)}}^{0.3333333333333333}\right)}^{3}\right) \]

   5. pow-pow 52.6%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(x \cdot x - 1\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}}^{3}\right) \]

   6. fma-neg 52.6%

      \[\leadsto \log \left(x + {\left({\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}^{3}\right) \]

   7. metadata-eval 52.6%

      \[\leadsto \log \left(x + {\left({\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}\right)}^{3}\right) \]

   8. metadata-eval 52.6%

      \[\leadsto \log \left(x + {\left({\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{\color{blue}{0.16666666666666666}}\right)}^{3}\right) \]

3. Applied egg-rr 52.6%

   \[\leadsto \log \left(x + \color{blue}{{\left({\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{0.16666666666666666}\right)}^{3}}\right) \]

4. Taylor expanded in x around inf 31.3%

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

   1. mul-1-neg 31.3%

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

   2. log-rec 31.3%

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

   3. remove-double-neg 31.3%

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

6. Simplified 31.3%

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

7. Final simplification 31.3%

   \[\leadsto \log x \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))