arccos

Percentage Accurate: 100.0% → 100.0%

Time: 8.1s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))

C

double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(sqrt(((1.0d0 - x) / (1.0d0 + x))))
end function

Java

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

Python

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (1.0 + x))))

Julia

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(1.0 + x)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))

C

double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(sqrt(((1.0d0 - x) / (1.0d0 + x))))
end function

Java

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

Python

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (1.0 + x))))

Julia

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(1.0 + x)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x \cdot x}}{1 + x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (/ (sqrt (- 1.0 (* x x))) (+ 1.0 x)))))

C

double code(double x) {
	return 2.0 * atan((sqrt((1.0 - (x * x))) / (1.0 + x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan((sqrt((1.0d0 - (x * x))) / (1.0d0 + x)))
end function

Java

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

Python

def code(x):
	return 2.0 * math.atan((math.sqrt((1.0 - (x * x))) / (1.0 + x)))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(sqrt(Float64(1.0 - Float64(x * x))) / Float64(1.0 + x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((sqrt((1.0 - (x * x))) / (1.0 + x)));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(N[Sqrt[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

3. Applied egg-rr 100.0%

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

   1. flip-- 100.0%

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

   2. frac-times 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. *-commutative 100.0%

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

   2. unpow2 100.0%

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

   3. associate-*r/ 100.0%

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

   4. *-rgt-identity 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

   \[\leadsto 2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x \cdot x}}{1 + x}\right) \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))

C

double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(sqrt(((1.0d0 - x) / (1.0d0 + x))))
end function

Java

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

Python

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (1.0 + x))))

Julia

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(1.0 + x)))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]

2. Final simplification 100.0%

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]

Alternative 3: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(((1.0d0 + ((x * x) * (-0.5d0))) / (1.0d0 + x)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(((1.0 + ((x * x) * -0.5)) / (1.0 + x)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

3. Applied egg-rr 100.0%

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

   1. flip-- 100.0%

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

   2. frac-times 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. *-commutative 100.0%

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

   2. unpow2 100.0%

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

   3. associate-*r/ 100.0%

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

   4. *-rgt-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 99.5%

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

    1. *-commutative 99.5%

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

    2. unpow2 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan((1.0d0 + ((x * (x * 0.5d0)) - x)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((1.0 + ((x * (x * 0.5)) - x)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

3. Applied egg-rr 100.0%

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

   1. flip-- 100.0%

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

   2. frac-times 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. *-commutative 100.0%

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

   2. unpow2 100.0%

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

   3. associate-*r/ 100.0%

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

   4. *-rgt-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 99.3%

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

    1. neg-mul-1 99.3%

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

    2. unsub-neg 99.3%

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

    3. *-commutative 99.3%

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

    4. unpow2 99.3%

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

    5. associate-*l* 99.3%

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

    \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(x \cdot \left(x \cdot 0.5\right) - x\right)\right) \]

Alternative 5: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (- 1.0 x))))

C

double code(double x) {
	return 2.0 * atan((1.0 - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan((1.0d0 - x))
end function

Java

public static double code(double x) {
	return 2.0 * Math.atan((1.0 - x));
}

Python

def code(x):
	return 2.0 * math.atan((1.0 - x))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((1.0 - x));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]

2. Taylor expanded in x around 0 98.8%

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

3. Taylor expanded in x around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. associate-+r+ 98.8%

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

   3. neg-mul-1 98.8%

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

   4. sub-neg 98.8%

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

   5. *-commutative 98.8%

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

   6. unpow2 98.8%

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

   7. associate-*l* 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in x around 0 98.8%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + -1 \cdot x\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 98.8%

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

   2. sub-neg 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

   \[\leadsto 2 \cdot \tan^{-1} \left(1 - x\right) \]

Alternative 6: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan 1.0)))

C

double code(double x) {
	return 2.0 * atan(1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(1.0d0)
end function

Java

public static double code(double x) {
	return 2.0 * Math.atan(1.0);
}

Python

def code(x):
	return 2.0 * math.atan(1.0)

Julia

function code(x)
	return Float64(2.0 * atan(1.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(1.0);
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

3. Applied egg-rr 100.0%

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

   1. flip-- 100.0%

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

   2. frac-times 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow2 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. *-commutative 100.0%

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

   2. unpow2 100.0%

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

   3. associate-*r/ 100.0%

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

   4. *-rgt-identity 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0 97.9%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{1} \]

10. Final simplification 97.9%

    \[\leadsto 2 \cdot \tan^{-1} 1 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))