arccos

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

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, 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. pow1/2 100.0%

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

   2. clear-num 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. 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 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: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + x \cdot \left(x \cdot -0.5\right)\\ 2 \cdot \tan^{-1} \left(\frac{1 - t_0 \cdot t_0}{1 + t_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (* x (* x -0.5)))))
   (* 2.0 (atan (/ (- 1.0 (* t_0 t_0)) (+ 1.0 t_0))))))

C

double code(double x) {
	double t_0 = x + (x * (x * -0.5));
	return 2.0 * atan(((1.0 - (t_0 * t_0)) / (1.0 + t_0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x + (x * (x * (-0.5d0)))
    code = 2.0d0 * atan(((1.0d0 - (t_0 * t_0)) / (1.0d0 + t_0)))
end function

Java

public static double code(double x) {
	double t_0 = x + (x * (x * -0.5));
	return 2.0 * Math.atan(((1.0 - (t_0 * t_0)) / (1.0 + t_0)));
}

Python

def code(x):
	t_0 = x + (x * (x * -0.5))
	return 2.0 * math.atan(((1.0 - (t_0 * t_0)) / (1.0 + t_0)))

Julia

function code(x)
	t_0 = Float64(x + Float64(x * Float64(x * -0.5)))
	return Float64(2.0 * atan(Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 + t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = x + (x * (x * -0.5));
	tmp = 2.0 * atan(((1.0 - (t_0 * t_0)) / (1.0 + t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[ArcTan[N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $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. associate-+r+ 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

   4. *-commutative 98.9%

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

   5. unpow2 98.9%

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

4. Simplified 98.9%

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

   1. associate-+l- 98.9%

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

   2. flip-- 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-commutative 98.9%

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

   5. cancel-sign-sub-inv 98.9%

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

   6. metadata-eval 98.9%

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

   7. *-commutative 98.9%

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

   8. cancel-sign-sub-inv 98.9%

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

   9. metadata-eval 98.9%

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

   10. *-commutative 98.9%

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

   11. cancel-sign-sub-inv 98.9%

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

   12. metadata-eval 98.9%

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

6. Applied egg-rr 98.9%

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

   1. associate-*r* 98.9%

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

   2. associate-*r* 98.9%

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

   3. associate-*r* 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[ArcTan[N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x + N[(x * N[(x * -0.5), $MachinePrecision]), $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. 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. associate-+r+ 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

   4. *-commutative 98.9%

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

   5. unpow2 98.9%

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

4. Simplified 98.9%

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

   1. associate-+l- 98.9%

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

   2. flip-- 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-commutative 98.9%

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

   5. cancel-sign-sub-inv 98.9%

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

   6. metadata-eval 98.9%

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

   7. *-commutative 98.9%

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

   8. cancel-sign-sub-inv 98.9%

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

   9. metadata-eval 98.9%

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

   10. *-commutative 98.9%

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

   11. cancel-sign-sub-inv 98.9%

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

   12. metadata-eval 98.9%

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

6. Applied egg-rr 98.9%

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

   1. associate-*r* 98.9%

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

   2. associate-*r* 98.9%

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

   3. associate-*r* 98.9%

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

8. Simplified 98.9%

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

9. Taylor expanded in x around 0 98.9%

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

    1. unpow2 98.9%

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 5: 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. 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. +-commutative 98.9%

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

   2. neg-mul-1 98.9%

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

   3. unsub-neg 98.9%

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

   4. *-commutative 98.9%

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

   5. unpow2 98.9%

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

   6. associate-*l* 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 6: 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. associate-+r+ 98.9%

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

   2. neg-mul-1 98.9%

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

   3. sub-neg 98.9%

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

   4. *-commutative 98.9%

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

   5. unpow2 98.9%

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

4. Simplified 98.9%

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

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 7: 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. neg-mul-1 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 8: 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. div-inv 100.0%

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

   2. sqrt-prod 100.0%

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

   3. pow1/2 100.0%

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

   4. inv-pow 100.0%

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

   5. pow-pow 100.0%

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

   6. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

5. Simplified 100.0%

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

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))))))