arccos

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 8

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, 0.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} \left(\sqrt[3]{{\left(\frac{1 + x}{1 - x}\right)}^{-1.5}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 2.0 (atan (cbrt (pow (/ (+ 1.0 x) (- 1.0 x)) -1.5)))))

C

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

Java

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

Julia

function code(x)
	return Float64(2.0 * atan(cbrt((Float64(Float64(1.0 + x) / Float64(1.0 - x)) ^ -1.5))))
end

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[Power[N[Power[N[(N[(1.0 + x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -1.5], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. pow1/2 99.9%

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

   2. metadata-eval 99.9%

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

   3. metadata-eval 99.9%

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

   4. pow-pow 100.0%

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

   5. clear-num 99.9%

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

   6. inv-pow 99.9%

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

   7. pow-pow 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

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

3. Applied egg-rr 99.9%

   \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left({\left({\left(\frac{1 + x}{1 - x}\right)}^{-1.5}\right)}^{0.3333333333333333}\right)} \]
4. Step-by-step derivation

   1. unpow1/3 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt[3]{{\left(\frac{1 + x}{1 - x}\right)}^{-1.5}}\right) \]

Alternative 2: 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 99.9%

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

   1. clear-num 99.9%

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

   2. sqrt-div 99.9%

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

   3. metadata-eval 99.9%

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

3. Applied egg-rr 99.9%

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

   1. flip-- 100.0%

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

   2. associate-/r/ 100.0%

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

   3. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. associate-*l/ 100.0%

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

   2. associate-/l* 100.0%

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

7. Simplified 100.0%

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

   1. add-sqr-sqrt 99.9%

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

   2. sqrt-unprod 100.0%

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

   3. frac-times 100.0%

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

   4. metadata-eval 100.0%

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

   5. add-sqr-sqrt 100.0%

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

9. Applied egg-rr 100.0%

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

    1. associate-/l* 100.0%

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

    2. *-lft-identity 100.0%

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

    3. associate-/l/ 99.9%

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

11. Simplified 99.9%

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

12. Taylor expanded in x around 0 100.0%

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

    1. associate-*l/ 100.0%

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

    2. *-lft-identity 100.0%

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

    3. unpow2 100.0%

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

14. Simplified 100.0%

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

15. Final simplification 100.0%

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

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 4: 99.2% 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 99.9%

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

2. Taylor expanded in x around 0 99.3%

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

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

4. Simplified 99.3%

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

5. 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: 99.2% accurate, 1.9× speedup

Math

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

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(Float64(1.0 + 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[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 99.3%

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

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

4. Simplified 99.3%

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

   1. associate-+r- 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 6: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. sqrt-div 99.9%

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

   3. metadata-eval 99.9%

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

3. Applied egg-rr 99.9%

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

4. Taylor expanded in x around 0 99.1%

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

5. Taylor expanded in x around 0 99.2%

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

   1. neg-mul-1 99.2%

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

   2. +-commutative 99.2%

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

   3. unsub-neg 99.2%

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

   4. unpow2 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 7: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. sqrt-div 99.9%

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

   3. metadata-eval 99.9%

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

3. Applied egg-rr 99.9%

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

4. Taylor expanded in x around 0 99.1%

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

5. Final simplification 99.1%

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

Alternative 8: 98.8% 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 99.9%

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

2. Taylor expanded in x around 0 99.1%

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

   1. neg-mul-1 99.1%

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

   2. sub-neg 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

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

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))