Hyperbolic arc-cosine

Percentage Accurate: 51.2% → 99.9%

Time: 7.8s

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: 51.2% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sqrt{x \cdot x + -1}\\ \mathbf{if}\;t\_0 \leq 4000000:\\ \;\;\;\;\log t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (sqrt (+ (* x x) -1.0)))))
   (if (<= t_0 4000000.0) (log t_0) (log (+ x x)))))

C

double code(double x) {
	double t_0 = x + sqrt(((x * x) + -1.0));
	double tmp;
	if (t_0 <= 4000000.0) {
		tmp = log(t_0);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + sqrt(((x * x) + (-1.0d0)))
    if (t_0 <= 4000000.0d0) then
        tmp = log(t_0)
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x + Math.sqrt(((x * x) + -1.0));
	double tmp;
	if (t_0 <= 4000000.0) {
		tmp = Math.log(t_0);
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + math.sqrt(((x * x) + -1.0))
	tmp = 0
	if t_0 <= 4000000.0:
		tmp = math.log(t_0)
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	t_0 = Float64(x + sqrt(Float64(Float64(x * x) + -1.0)))
	tmp = 0.0
	if (t_0 <= 4000000.0)
		tmp = log(t_0);
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x + sqrt(((x * x) + -1.0));
	tmp = 0.0;
	if (t_0 <= 4000000.0)
		tmp = log(t_0);
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4000000.0], N[Log[t$95$0], $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64)))) < 4e6

   1. Initial program 99.7%

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

   if 4e6 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 48.8%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 4000000:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. 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 50.8%

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

3. Taylor expanded in x around inf 98.3%

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

Alternative 3: 13.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log 1.625 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log 1.625))

C

double code(double x) {
	return log(1.625);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(1.625d0)
end function

Java

public static double code(double x) {
	return Math.log(1.625);
}

Python

def code(x):
	return math.log(1.625)

Julia

function code(x)
	return log(1.625)
end

MATLAB

function tmp = code(x)
	tmp = log(1.625);
end

Wolfram

code[x_] := N[Log[1.625], $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. pow2 50.8%

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

   2. add-cube-cbrt 50.8%

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

   3. pow3 50.8%

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

   4. pow-pow 50.8%

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

   5. metadata-eval 50.8%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt[3]{x}\right)}^{\color{blue}{6}} - 1}\right) \]

4. Applied egg-rr 50.8%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt[3]{x}\right)}^{6}} - 1}\right) \]

5. Taylor expanded in x around -inf 99.6%

   \[\leadsto \log \color{blue}{\left(-1 \cdot \left(x \cdot \left(\left(-1 \cdot \frac{0.5 \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{3}} + 0.125 \cdot \frac{1}{{x}^{2} \cdot {\left(\sqrt[3]{-1}\right)}^{9}}}{{x}^{2}} + {\left(\sqrt[3]{-1}\right)}^{3}\right) - 1\right)\right)\right)} \]

7. Simplified 13.8%

   \[\leadsto \log \color{blue}{1.625} \]
8. Add Preprocessing

Alternative 4: 5.5% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.075 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.075))

C

double code(double x) {
	return x * 0.075;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.075d0
end function

Java

public static double code(double x) {
	return x * 0.075;
}

Python

def code(x):
	return x * 0.075

Julia

function code(x)
	return Float64(x * 0.075)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.075;
end

Wolfram

code[x_] := N[(x * 0.075), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

   1. pow2 50.8%

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

   2. add-cube-cbrt 50.8%

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

   3. pow3 50.8%

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

   4. pow-pow 50.8%

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

   5. metadata-eval 50.8%

      \[\leadsto \log \left(x + \sqrt{{\left(\sqrt[3]{x}\right)}^{\color{blue}{6}} - 1}\right) \]

4. Applied egg-rr 50.8%

   \[\leadsto \log \left(x + \sqrt{\color{blue}{{\left(\sqrt[3]{x}\right)}^{6}} - 1}\right) \]

5. Taylor expanded in x around 0 0.0%

   \[\leadsto \color{blue}{\log \left(\sqrt{-1}\right) + x \cdot \left({x}^{2} \cdot \left(0.075 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{5}} - 0.16666666666666666 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{3}}\right) + \frac{1}{\sqrt{-1}}\right)} \]

7. Simplified 5.5%

   \[\leadsto \color{blue}{-2 + x \cdot 0.075} \]

8. Taylor expanded in x around inf 5.5%

   \[\leadsto \color{blue}{x \cdot \left(0.075 - 2 \cdot \frac{1}{x}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 5.5%

      \[\leadsto x \cdot \left(0.075 - \color{blue}{\frac{2 \cdot 1}{x}}\right) \]

   2. metadata-eval 5.5%

      \[\leadsto x \cdot \left(0.075 - \frac{\color{blue}{2}}{x}\right) \]

10. Simplified 5.5%

    \[\leadsto \color{blue}{x \cdot \left(0.075 - \frac{2}{x}\right)} \]

11. Taylor expanded in x around inf 5.6%

    \[\leadsto x \cdot \color{blue}{0.075} \]
12. Add Preprocessing

Alternative 5: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.6666666666666666)

C

double code(double x) {
	return -0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.6666666666666666d0
end function

Java

public static double code(double x) {
	return -0.6666666666666666;
}

Python

def code(x):
	return -0.6666666666666666

Julia

function code(x)
	return -0.6666666666666666
end

MATLAB

function tmp = code(x)
	tmp = -0.6666666666666666;
end

Wolfram

code[x_] := -0.6666666666666666

Derivation

1. Initial program 50.8%

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

   1. add-cbrt-cube 33.0%

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

   2. pow1/3 32.8%

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

   3. log-pow 32.8%

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

   4. pow3 32.8%

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

   5. log-pow 50.6%

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

   6. fma-neg 50.6%

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

   7. metadata-eval 50.6%

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

4. Applied egg-rr 50.6%

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

5. Taylor expanded in x around 0 0.0%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt{-1}\right) + x \cdot \left({x}^{2} \cdot \left(0.225 \cdot \frac{{x}^{2}}{{\left(\sqrt{-1}\right)}^{5}} - 0.5 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{3}}\right) + 3 \cdot \frac{1}{\sqrt{-1}}\right)\right)} \]

7. Simplified 5.5%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(-2 + x \cdot 0.725\right)} \]

8. Taylor expanded in x around 0 1.6%

   \[\leadsto \color{blue}{-0.6666666666666666} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))