Hyperbolic arc-cosine

Percentage Accurate: 51.9% → 99.8%

Time: 6.6s

Alternatives: 6

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: 51.9% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sqrt{x \cdot x + -1}\\ \mathbf{if}\;t\_0 \leq 10^{+97}:\\ \;\;\;\;\log t\_0\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (sqrt (+ (* x x) -1.0)))))
   (if (<= t_0 1e+97) (log t_0) (+ (log 2.0) (log x)))))

C

double code(double x) {
	double t_0 = x + sqrt(((x * x) + -1.0));
	double tmp;
	if (t_0 <= 1e+97) {
		tmp = log(t_0);
	} else {
		tmp = log(2.0) + log(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 <= 1d+97) then
        tmp = log(t_0)
    else
        tmp = log(2.0d0) + log(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 <= 1e+97) {
		tmp = Math.log(t_0);
	} else {
		tmp = Math.log(2.0) + Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + math.sqrt(((x * x) + -1.0))
	tmp = 0
	if t_0 <= 1e+97:
		tmp = math.log(t_0)
	else:
		tmp = math.log(2.0) + math.log(x)
	return tmp

Julia

function code(x)
	t_0 = Float64(x + sqrt(Float64(Float64(x * x) + -1.0)))
	tmp = 0.0
	if (t_0 <= 1e+97)
		tmp = log(t_0);
	else
		tmp = Float64(log(2.0) + log(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 <= 1e+97)
		tmp = log(t_0);
	else
		tmp = log(2.0) + log(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, 1e+97], N[Log[t$95$0], $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64)))) < 1.0000000000000001e97

   1. Initial program 100.0%

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

   if 1.0000000000000001e97 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 31.0%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. log-rec 99.7%

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

      3. remove-double-neg 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 10^{+97}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% 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 51.2%

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

   1. pow1/2 51.2%

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

   2. difference-of-sqr-1 51.2%

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

   3. unpow-prod-down 99.6%

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

   4. sub-neg 99.6%

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

   5. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

   1. unpow1/2 99.6%

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

   2. unpow1/2 99.6%

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

6. Simplified 99.6%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log 2 + \log x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (log 2.0) (log x)))

C

double code(double x) {
	return log(2.0) + log(x);
}

Fortran

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

Java

public static double code(double x) {
	return Math.log(2.0) + Math.log(x);
}

Python

def code(x):
	return math.log(2.0) + math.log(x)

Julia

function code(x)
	return Float64(log(2.0) + log(x))
end

MATLAB

function tmp = code(x)
	tmp = log(2.0) + log(x);
end

Wolfram

code[x_] := N[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around inf 97.9%

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

   1. mul-1-neg 97.9%

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

   2. log-rec 97.9%

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

   3. remove-double-neg 97.9%

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

5. Simplified 97.9%

   \[\leadsto \color{blue}{\log 2 + \log x} \]
6. Add Preprocessing

Alternative 4: 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 51.2%

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

3. Taylor expanded in x around inf 97.7%

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

Alternative 5: 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 51.2%

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

3. Taylor expanded in x around inf 97.7%

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

4. Taylor expanded in x around 0 97.9%

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

6. Simplified 31.4%

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

Alternative 6: 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 51.2%

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

   1. pow1/2 51.2%

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

   2. difference-of-sqr-1 51.2%

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

   3. unpow-prod-down 99.6%

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

   4. sub-neg 99.6%

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

   5. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

   1. unpow1/2 99.6%

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

   2. unpow1/2 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in x around 0 0.0%

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

9. Simplified 1.1%

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

10. Taylor expanded in x around 0 1.6%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))