arccos

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
2. Add Preprocessing
3. 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. div-inv 100.0%

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

4. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 3: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return Float64(2.0 * atan(Float64(1.0 + Float64(x * Float64(-1.0 + Float64(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[(1.0 + N[(x * N[(-1.0 + 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. Add Preprocessing

3. Taylor expanded in x around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. unpow2 99.3%

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

   4. associate-*l* 99.3%

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

   5. distribute-lft-out 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 4: 99.1% 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 100.0%

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

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0 99.0%

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

   1. +-commutative 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 5: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 99.0%

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

   1. neg-mul-1 99.0%

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

   2. sub-neg 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 6: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \tan^{-1} 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. unpow2 99.3%

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

   4. associate-*l* 99.3%

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

   5. distribute-lft-out 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 97.4%

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

7. Final simplification 97.4%

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

Reproduce

herbie shell --seed 2024018 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))