arccos

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 3

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

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

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 2: 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 98.6%

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

   1. +-commutative 98.6%

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

   2. neg-mul-1 98.6%

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

   3. unsub-neg 98.6%

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

   4. *-commutative 98.6%

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

4. Simplified 98.6%

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

5. Taylor expanded in x around 0 98.6%

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

   1. associate--l+ 98.6%

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

   2. unpow2 98.6%

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

   3. associate-*r* 98.6%

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

   4. *-lft-identity 98.6%

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

   5. distribute-rgt-out-- 98.6%

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

   6. fma-neg 98.6%

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

   7. metadata-eval 98.6%

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

7. Simplified 98.6%

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

   1. fma-udef 98.6%

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

   2. distribute-rgt-in 98.6%

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

   3. *-commutative 98.6%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 3: 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 98.1%

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

   1. neg-mul-1 98.1%

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

   2. sub-neg 98.1%

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

4. Simplified 98.1%

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

5. Final simplification 98.1%

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

Reproduce

herbie shell --seed 2023332 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))