Average Error: 0.0 → 0.0
Time: 5.0s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{t_0}}{t_0}}\right) \end{array} \]
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (* 2.0 (atan (sqrt (/ (/ (- 1.0 x) t_0) t_0))))))
double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}
↓
double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return 2.0 * atan(sqrt((((1.0 - x) / t_0) / t_0)));
}

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

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

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

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

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

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

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Applied add-sqr-sqrt_binary640.0

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

  3. Applied associate-/r*_binary640.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))