arccos

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 9

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. sqrt-div 100.0%

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

   2. flip-- 100.0%

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

   3. sqrt-div 99.9%

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

   4. associate-/l/ 100.0%

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

   5. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. rem-square-sqrt 100.0%

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

5. Simplified 100.0%

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

6. 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. metadata-eval 100.0%

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

   4. sqrt-pow1 100.0%

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

   5. metadata-eval 100.0%

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

   6. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 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 4: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. flip-- 100.0%

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

   3. sqrt-div 99.9%

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

   4. associate-/l/ 100.0%

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

   5. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. rem-square-sqrt 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

8. Simplified 99.3%

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

   1. frac-2neg 99.3%

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

   2. div-inv 99.3%

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

   3. *-commutative 99.3%

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

   4. distribute-neg-in 99.3%

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

   5. unsub-neg 99.3%

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

   6. metadata-eval 99.3%

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

   7. distribute-neg-in 99.3%

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

   8. metadata-eval 99.3%

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

   9. distribute-rgt-neg-in 99.3%

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

   10. distribute-rgt-neg-in 99.3%

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

   11. metadata-eval 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 5: 99.5% accurate, 1.8× speedup

Math

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

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(x * Float64(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[(x * N[(x * -0.5), $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. sqrt-div 100.0%

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

   2. flip-- 100.0%

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

   3. sqrt-div 99.9%

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

   4. associate-/l/ 100.0%

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

   5. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. rem-square-sqrt 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 6: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(1 + \left(x \cdot \left(x \cdot 0.625\right) - x\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (atan (+ 1.0 (- (* x (* x 0.625)) x)))))

C

double code(double x) {
	return 2.0 * atan((1.0 + ((x * (x * 0.625)) - x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 * atan((1.0d0 + ((x * (x * 0.625d0)) - x)))
end function

Java

public static double code(double x) {
	return 2.0 * Math.atan((1.0 + ((x * (x * 0.625)) - x)));
}

Python

def code(x):
	return 2.0 * math.atan((1.0 + ((x * (x * 0.625)) - x)))

Julia

function code(x)
	return Float64(2.0 * atan(Float64(1.0 + Float64(Float64(x * Float64(x * 0.625)) - x))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * atan((1.0 + ((x * (x * 0.625)) - x)));
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(1.0 + N[(N[(x * N[(x * 0.625), $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. div-inv 100.0%

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

   2. *-commutative 100.0%

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

   3. sqrt-prod 100.0%

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

   4. sqrt-div 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow1/2 100.0%

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

   7. pow-flip 100.0%

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

   8. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in x around 0 98.6%

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

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

7. Taylor expanded in x around 0 98.6%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + \left(-1 \cdot x + 0.625 \cdot {x}^{2}\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative 98.6%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \color{blue}{\left(0.625 \cdot {x}^{2} + -1 \cdot x\right)}\right) \]

   2. mul-1-neg 98.6%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(0.625 \cdot {x}^{2} + \color{blue}{\left(-x\right)}\right)\right) \]

   3. unsub-neg 98.6%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \color{blue}{\left(0.625 \cdot {x}^{2} - x\right)}\right) \]

   4. *-commutative 98.6%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\color{blue}{{x}^{2} \cdot 0.625} - x\right)\right) \]

   5. unpow2 98.6%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.625 - x\right)\right) \]

   6. associate-*l* 98.6%

      \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(\color{blue}{x \cdot \left(x \cdot 0.625\right)} - x\right)\right) \]

9. Simplified 98.6%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + \left(x \cdot \left(x \cdot 0.625\right) - x\right)\right)} \]

10. Final simplification 98.6%

    \[\leadsto 2 \cdot \tan^{-1} \left(1 + \left(x \cdot \left(x \cdot 0.625\right) - x\right)\right) \]

Alternative 7: 99.3% 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 98.9%

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

   1. mul-1-neg 98.9%

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

   2. associate-+r+ 98.9%

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

   3. sub-neg 98.9%

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

   4. unpow2 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 8: 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 98.5%

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

   1. mul-1-neg 98.5%

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

   2. sub-neg 98.5%

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

4. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 9: 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. sqrt-div 100.0%

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

   2. flip-- 100.0%

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

   3. sqrt-div 99.9%

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

   4. associate-/l/ 100.0%

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

   5. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. rem-square-sqrt 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around 0 97.4%

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

7. Final simplification 97.4%

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))