Hyperbolic arc-cosine

Percentage Accurate: 51.2% → 99.9%

Time: 5.4s

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: 51.2% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 20000000:\\ \;\;\;\;\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x (sqrt (+ (* x x) -1.0))) 20000000.0)
   (log1p (+ (+ x (sqrt (fma x x -1.0))) -1.0))
   (log (+ x x))))

C

double code(double x) {
	double tmp;
	if ((x + sqrt(((x * x) + -1.0))) <= 20000000.0) {
		tmp = log1p(((x + sqrt(fma(x, x, -1.0))) + -1.0));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + sqrt(Float64(Float64(x * x) + -1.0))) <= 20000000.0)
		tmp = log1p(Float64(Float64(x + sqrt(fma(x, x, -1.0))) + -1.0));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 20000000.0], N[Log[1 + N[(N[(x + N[Sqrt[N[(x * x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $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)))) < 2e7

   1. Initial program 99.7%

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

      1. log1p-expm1-u 99.8%

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

      2. expm1-undefine 99.8%

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

      3. add-exp-log 99.8%

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

      4. fma-neg 99.8%

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

      5. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

   if 2e7 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 50.0%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 20000000:\\ \;\;\;\;\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sqrt{x \cdot x + -1}\\ \mathbf{if}\;t\_0 \leq 20000000:\\ \;\;\;\;\log t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (sqrt (+ (* x x) -1.0)))))
   (if (<= t_0 20000000.0) (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 <= 20000000.0) {
		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 <= 20000000.0d0) 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 <= 20000000.0) {
		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 <= 20000000.0:
		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 <= 20000000.0)
		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 <= 20000000.0)
		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, 20000000.0], 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)))) < 2e7

   1. Initial program 99.7%

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

   if 2e7 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 50.0%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 20000000:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% 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 51.9%

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

3. Taylor expanded in x around inf 98.1%

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

4. Final simplification 98.1%

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

Reproduce

herbie shell --seed 2024075 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))