Hyperbolic arc-cosine

Percentage Accurate: 50.8% → 99.4%

Time: 7.6s

Alternatives: 6

Speedup: 1.0×

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: 50.8% 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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.4%

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

3. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

5. Simplified 99.2%

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

Alternative 2: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sqrt{x \cdot x + -1}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\log t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (sqrt (+ (* x x) -1.0)))))
   (if (<= t_0 5e+153) (log t_0) (+ (/ 1.0 x) (log x)))))

C

double code(double x) {
	double t_0 = x + sqrt(((x * x) + -1.0));
	double tmp;
	if (t_0 <= 5e+153) {
		tmp = log(t_0);
	} else {
		tmp = (1.0 / x) + 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 <= 5d+153) then
        tmp = log(t_0)
    else
        tmp = (1.0d0 / x) + 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 <= 5e+153) {
		tmp = Math.log(t_0);
	} else {
		tmp = (1.0 / x) + Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + math.sqrt(((x * x) + -1.0))
	tmp = 0
	if t_0 <= 5e+153:
		tmp = math.log(t_0)
	else:
		tmp = (1.0 / x) + 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 <= 5e+153)
		tmp = log(t_0);
	else
		tmp = Float64(Float64(1.0 / x) + 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 <= 5e+153)
		tmp = log(t_0);
	else
		tmp = (1.0 / x) + 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, 5e+153], N[Log[t$95$0], $MachinePrecision], N[(N[(1.0 / x), $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)))) < 5.00000000000000018e153

   1. Initial program 100.0%

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

   if 5.00000000000000018e153 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 3.1%

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

   3. Taylor expanded in x around 0 0.0%

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

   5. Applied egg-rr 33.0%

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

   6. Taylor expanded in x around inf 33.0%

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

      1. +-commutative 33.0%

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

      2. mul-1-neg 33.0%

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

      3. log-rec 33.0%

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

      4. remove-double-neg 33.0%

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

   8. Simplified 33.0%

      \[\leadsto \color{blue}{\frac{1}{x} + \log x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \log x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.4%

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

3. Taylor expanded in x around 0 0.0%

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

5. Applied egg-rr 31.5%

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

6. Taylor expanded in x around inf 31.5%

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

   1. +-commutative 31.5%

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

   2. mul-1-neg 31.5%

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

   3. log-rec 31.5%

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

   4. remove-double-neg 31.5%

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

8. Simplified 31.5%

   \[\leadsto \color{blue}{\frac{1}{x} + \log x} \]
9. Add Preprocessing

Alternative 4: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p x))

C

double code(double x) {
	return log1p(x);
}

Java

public static double code(double x) {
	return Math.log1p(x);
}

Python

def code(x):
	return math.log1p(x)

Julia

function code(x)
	return log1p(x)
end

Wolfram

code[x_] := N[Log[1 + x], $MachinePrecision]

Derivation

1. Initial program 47.4%

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

3. Taylor expanded in x around 0 0.0%

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

5. Applied egg-rr 31.5%

   \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(x\right)} \]
6. Step-by-step derivation

   1. +-lft-identity 31.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]

7. Simplified 31.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
8. Add Preprocessing

Alternative 5: 3.9% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* x (- (* x 0.3333333333333333) 0.5)))))

C

double code(double x) {
	return x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * ((x * 0.3333333333333333d0) - 0.5d0)))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
}

Python

def code(x):
	return x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * 0.3333333333333333) - 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.4%

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

3. Taylor expanded in x around 0 0.0%

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

5. Applied egg-rr 31.5%

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

6. Taylor expanded in x around 0 4.0%

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

7. Final simplification 4.0%

   \[\leadsto x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right) \]
8. Add Preprocessing

Alternative 6: 0.9% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (-0.5d0)))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * -0.5));
}

Python

def code(x):
	return x * (1.0 + (x * -0.5))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * -0.5));
end

Wolfram

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

Derivation

1. Initial program 47.4%

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

3. Taylor expanded in x around 0 0.0%

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

5. Applied egg-rr 31.5%

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

6. Taylor expanded in x around 0 0.9%

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

   1. *-commutative 0.9%

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

8. Simplified 0.9%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot -0.5\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))