arccos

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(1.0 / N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(1.0 - x), $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. 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. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

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

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

FPCore

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

C

double code(double x) {
	return 2.0 * atan((1.0 + (((x * x) * (0.5 + (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 * (-0.5d0)))) - x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(1.0 + N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(x * -0.5), $MachinePrecision]), $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. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in x around 0 99.7%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + \left(-1 \cdot x + \left(-0.5 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right)\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \color{blue}{\left(\left(-0.5 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right) + -1 \cdot x\right)}\right) \]

   2. neg-mul-1 99.7%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\left(-0.5 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right) + \color{blue}{\left(-x\right)}\right)\right) \]

   3. unsub-neg 99.7%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \color{blue}{\left(\left(-0.5 \cdot {x}^{3} + 0.5 \cdot {x}^{2}\right) - x\right)}\right) \]

   4. +-commutative 99.7%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\color{blue}{\left(0.5 \cdot {x}^{2} + -0.5 \cdot {x}^{3}\right)} - x\right)\right) \]

   5. *-commutative 99.7%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\left(\color{blue}{{x}^{2} \cdot 0.5} + -0.5 \cdot {x}^{3}\right) - x\right)\right) \]

   6. *-commutative 99.7%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\left({x}^{2} \cdot 0.5 + \color{blue}{{x}^{3} \cdot -0.5}\right) - x\right)\right) \]

   7. unpow3 99.7%

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

   8. unpow2 99.7%

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

   9. associate-*l* 99.7%

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

   10. distribute-lft-out 99.7%

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

   11. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(1.0 / N[(N[(1.0 + x), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $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. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in x around 0 99.7%

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

   1. associate-+r+ 99.7%

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

   2. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(N[(1.0 - x), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $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 99.7%

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

   1. associate-+r+ 99.7%

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

   2. neg-mul-1 99.7%

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

   3. sub-neg 99.7%

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

   4. *-commutative 99.7%

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

   5. unpow2 99.7%

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

4. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 6: 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. Taylor expanded in x around 0 99.6%

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

   1. neg-mul-1 99.6%

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

   2. sub-neg 99.6%

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

4. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 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. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in x around 0 99.6%

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

5. Taylor expanded in x around 0 99.0%

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

6. Final simplification 99.0%

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

Reproduce

herbie shell --seed 2023285 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))