arccos

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 5

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

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

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left({\left(1 - x \cdot 0.6666666666666666\right)}^{1.5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (pow (- 1.0 (* x 0.6666666666666666)) 1.5))))

C

double code(double x) {
	return 2.0 * atan(pow((1.0 - (x * 0.6666666666666666)), 1.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan(((1.0d0 - (x * 0.6666666666666666d0)) ** 1.5d0))
end function

Java

public static double code(double x) {
	return 2.0 * Math.atan(Math.pow((1.0 - (x * 0.6666666666666666)), 1.5));
}

Python

def code(x):
	return 2.0 * math.atan(math.pow((1.0 - (x * 0.6666666666666666)), 1.5))

Julia

function code(x)
	return Float64(2.0 * atan((Float64(1.0 - Float64(x * 0.6666666666666666)) ^ 1.5)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan(((1.0 - (x * 0.6666666666666666)) ^ 1.5));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[Power[N[(1.0 - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-cube-cbrt 100.0%

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

   2. pow3 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.1%

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\color{blue}{\left(1 + -0.6666666666666666 \cdot x\right)}}^{3}}\right) \]
6. Step-by-step derivation

   1. *-commutative 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\left(1 + \color{blue}{x \cdot -0.6666666666666666}\right)}^{3}}\right) \]

7. Simplified 99.1%

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\color{blue}{\left(1 + x \cdot -0.6666666666666666\right)}}^{3}}\right) \]

8. Taylor expanded in x around 0 99.1%

   \[\leadsto 2 \cdot \color{blue}{\tan^{-1} \left(\sqrt{{\left(1 + -0.6666666666666666 \cdot x\right)}^{3}}\right)} \]
9. Step-by-step derivation

   1. +-commutative 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\color{blue}{\left(-0.6666666666666666 \cdot x + 1\right)}}^{3}}\right) \]

   2. *-commutative 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\left(\color{blue}{x \cdot -0.6666666666666666} + 1\right)}^{3}}\right) \]

   3. fma-undefine 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\color{blue}{\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}}^{3}}\right) \]

   4. metadata-eval 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{\color{blue}{\left(2 \cdot 1.5\right)}}}\right) \]

   5. pow-sqr 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5} \cdot {\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}}}\right) \]

   6. rem-sqrt-square 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\left|{\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right|\right)} \]

   7. unpow1 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\left|\color{blue}{{\left({\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right)}^{1}}\right|\right) \]

   8. sqr-pow 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\left|\color{blue}{{\left({\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right)}^{\left(\frac{1}{2}\right)}}\right|\right) \]

   9. fabs-sqr 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left({\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right)}^{\left(\frac{1}{2}\right)}\right)} \]

   10. sqr-pow 99.1%

       \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left({\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right)}^{1}\right)} \]

   11. unpow1 99.1%

       \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right)} \]

10. Simplified 99.1%

    \[\leadsto 2 \cdot \color{blue}{\tan^{-1} \left({\left(\mathsf{fma}\left(x, -0.6666666666666666, 1\right)\right)}^{1.5}\right)} \]

11. Taylor expanded in x around inf 99.1%

    \[\leadsto 2 \cdot \color{blue}{\tan^{-1} \left({\left(1 - 0.6666666666666666 \cdot x\right)}^{1.5}\right)} \]

12. Final simplification 99.1%

    \[\leadsto 2 \cdot \tan^{-1} \left({\left(1 - x \cdot 0.6666666666666666\right)}^{1.5}\right) \]
13. Add Preprocessing

Alternative 3: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(1 + x \cdot \left(-1 + x \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (+ 1.0 (* x (+ -1.0 (* x 0.16666666666666666)))))))

C

double code(double x) {
	return 2.0 * atan((1.0 + (x * (-1.0 + (x * 0.16666666666666666)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan((1.0d0 + (x * ((-1.0d0) + (x * 0.16666666666666666d0)))))
end function

Java

public static double code(double x) {
	return 2.0 * Math.atan((1.0 + (x * (-1.0 + (x * 0.16666666666666666)))));
}

Python

def code(x):
	return 2.0 * math.atan((1.0 + (x * (-1.0 + (x * 0.16666666666666666)))))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * 0.16666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((1.0 + (x * (-1.0 + (x * 0.16666666666666666)))));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(1.0 + N[(x * N[(-1.0 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. add-cube-cbrt 100.0%

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

   2. pow3 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.1%

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\color{blue}{\left(1 + -0.6666666666666666 \cdot x\right)}}^{3}}\right) \]
6. Step-by-step derivation

   1. *-commutative 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\left(1 + \color{blue}{x \cdot -0.6666666666666666}\right)}^{3}}\right) \]

7. Simplified 99.1%

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{{\color{blue}{\left(1 + x \cdot -0.6666666666666666\right)}}^{3}}\right) \]

8. Taylor expanded in x around 0 99.1%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + \left(-1 \cdot x + 0.16666666666666666 \cdot {x}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. unpow2 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(-1 \cdot x + 0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   2. associate-*r* 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(-1 \cdot x + \color{blue}{\left(0.16666666666666666 \cdot x\right) \cdot x}\right)\right) \]

   3. distribute-rgt-out 99.1%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \color{blue}{x \cdot \left(-1 + 0.16666666666666666 \cdot x\right)}\right) \]

10. Simplified 99.1%

    \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + x \cdot \left(-1 + 0.16666666666666666 \cdot x\right)\right)} \]

11. Final simplification 99.1%

    \[\leadsto 2 \cdot \tan^{-1} \left(1 + x \cdot \left(-1 + x \cdot 0.16666666666666666\right)\right) \]
12. Add Preprocessing

Alternative 4: 99.0% 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. Add Preprocessing

3. Taylor expanded in x around 0 99.0%

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

   1. *-commutative 99.0%

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

5. Simplified 99.0%

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

   1. add-cbrt-cube 99.0%

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

   2. pow1/3 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left(\left(\sqrt{1 + x \cdot -2} \cdot \sqrt{1 + x \cdot -2}\right) \cdot \sqrt{1 + x \cdot -2}\right)}^{0.3333333333333333}\right)} \]

   3. add-sqr-sqrt 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left(\color{blue}{\left(1 + x \cdot -2\right)} \cdot \sqrt{1 + x \cdot -2}\right)}^{0.3333333333333333}\right) \]

   4. pow1 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left(\color{blue}{{\left(1 + x \cdot -2\right)}^{1}} \cdot \sqrt{1 + x \cdot -2}\right)}^{0.3333333333333333}\right) \]

   5. pow1/2 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left({\left(1 + x \cdot -2\right)}^{1} \cdot \color{blue}{{\left(1 + x \cdot -2\right)}^{0.5}}\right)}^{0.3333333333333333}\right) \]

   6. pow-prod-up 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\color{blue}{\left({\left(1 + x \cdot -2\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right) \]

   7. +-commutative 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left({\color{blue}{\left(x \cdot -2 + 1\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}\right) \]

   8. fma-define 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left({\color{blue}{\left(\mathsf{fma}\left(x, -2, 1\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}\right) \]

   9. metadata-eval 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left({\left(\mathsf{fma}\left(x, -2, 1\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right) \]

7. Applied egg-rr 99.0%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left({\left(\mathsf{fma}\left(x, -2, 1\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]
8. Step-by-step derivation

   1. unpow1/3 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\sqrt[3]{{\left(\mathsf{fma}\left(x, -2, 1\right)\right)}^{1.5}}\right)} \]

9. Simplified 99.0%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\sqrt[3]{{\left(\mathsf{fma}\left(x, -2, 1\right)\right)}^{1.5}}\right)} \]

10. Taylor expanded in x around 0 99.0%

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

    1. mul-1-neg 99.0%

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

    2. unsub-neg 99.0%

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

12. Simplified 99.0%

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

13. Final simplification 99.0%

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

Alternative 5: 98.0% 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. Add Preprocessing

3. Taylor expanded in x around 0 99.0%

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

   1. *-commutative 99.0%

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

5. Simplified 99.0%

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

   1. add-cbrt-cube 99.0%

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

   2. pow1/3 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left(\left(\sqrt{1 + x \cdot -2} \cdot \sqrt{1 + x \cdot -2}\right) \cdot \sqrt{1 + x \cdot -2}\right)}^{0.3333333333333333}\right)} \]

   3. add-sqr-sqrt 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left(\color{blue}{\left(1 + x \cdot -2\right)} \cdot \sqrt{1 + x \cdot -2}\right)}^{0.3333333333333333}\right) \]

   4. pow1 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left(\color{blue}{{\left(1 + x \cdot -2\right)}^{1}} \cdot \sqrt{1 + x \cdot -2}\right)}^{0.3333333333333333}\right) \]

   5. pow1/2 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left({\left(1 + x \cdot -2\right)}^{1} \cdot \color{blue}{{\left(1 + x \cdot -2\right)}^{0.5}}\right)}^{0.3333333333333333}\right) \]

   6. pow-prod-up 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\color{blue}{\left({\left(1 + x \cdot -2\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right) \]

   7. +-commutative 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left({\color{blue}{\left(x \cdot -2 + 1\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}\right) \]

   8. fma-define 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left({\color{blue}{\left(\mathsf{fma}\left(x, -2, 1\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333}\right) \]

   9. metadata-eval 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \left({\left({\left(\mathsf{fma}\left(x, -2, 1\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right) \]

7. Applied egg-rr 99.0%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left({\left(\mathsf{fma}\left(x, -2, 1\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)} \]
8. Step-by-step derivation

   1. unpow1/3 99.0%

      \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\sqrt[3]{{\left(\mathsf{fma}\left(x, -2, 1\right)\right)}^{1.5}}\right)} \]

9. Simplified 99.0%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(\sqrt[3]{{\left(\mathsf{fma}\left(x, -2, 1\right)\right)}^{1.5}}\right)} \]

10. Taylor expanded in x around 0 98.2%

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

11. Final simplification 98.2%

    \[\leadsto 2 \cdot \tan^{-1} 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))