arccos

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. div-sub 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. clear-num 100.0%

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

   2. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in x around 0 98.7%

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

5. Taylor expanded in x around 0 98.7%

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

   1. neg-mul-1 98.7%

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

   2. associate-+r+ 98.7%

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

   3. sub-neg 98.7%

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

7. Simplified 98.7%

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

   1. sub-neg 98.7%

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

   2. associate-+l+ 98.7%

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

   3. neg-mul-1 98.7%

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

   4. unpow2 98.7%

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

   5. distribute-rgt-in 98.7%

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

   6. +-commutative 98.7%

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

   7. distribute-rgt-in 98.7%

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

   8. unpow2 98.7%

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

   9. metadata-eval 98.7%

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

   10. cancel-sign-sub-inv 98.7%

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

   11. unpow2 98.7%

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

   12. +-commutative 98.7%

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

   13. unpow2 98.7%

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

   14. cancel-sign-sub-inv 98.7%

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

   15. unpow2 98.7%

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

   16. metadata-eval 98.7%

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

   17. distribute-rgt-in 98.7%

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

   18. flip-+ 98.7%

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

   19. metadata-eval 98.7%

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

9. Applied egg-rr 98.7%

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

10. Final simplification 98.7%

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

Alternative 4: 98.9% 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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. sqrt-div 100.0%

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

   3. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Taylor expanded in x around 0 98.7%

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

5. Final simplification 98.7%

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

Alternative 5: 98.9% 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 98.7%

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

   1. neg-mul-1 98.7%

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

   2. sub-neg 98.7%

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

4. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 6: 97.8% 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. Taylor expanded in x around 0 97.7%

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

3. Final simplification 97.7%

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

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))