?

Average Error: 0.0 → 0.0
Time: 6.6s
Precision: binary64
Cost: 14016

?

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

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 = 1.0d0 - (x * x)
    code = 2.0d0 * atan(((1.0d0 / t_0) * (sqrt(t_0) * (1.0d0 - x))))
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 = 1.0 - (x * x);
	return 2.0 * Math.atan(((1.0 / t_0) * (Math.sqrt(t_0) * (1.0 - x))));
}

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

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

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end
↓
function tmp = code(x)
	t_0 = 1.0 - (x * x);
	tmp = 2.0 * atan(((1.0 / t_0) * (sqrt(t_0) * (1.0 - x))));
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[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[ArcTan[N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

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

Error?

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 egg-rr0.0

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost13376
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
Alternative 2
Error0.3
Cost7488
\[2 \cdot \tan^{-1} \left(\frac{1 + \left(1 + \left(-1 + x \cdot \left(x \cdot -0.5\right)\right)\right)}{1 + x}\right) \]
Alternative 3
Error0.3
Cost7360
\[2 \cdot \tan^{-1} \left(1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot -0.5 + 0.5\right) - x\right)\right) \]
Alternative 4
Error0.3
Cost7232
\[2 \cdot \tan^{-1} \left(\frac{1 - \left(x \cdot x\right) \cdot 0.5}{1 + x}\right) \]
Alternative 5
Error0.5
Cost7104
\[2 \cdot \tan^{-1} \left(1 - \left(x - x \cdot \left(x \cdot 0.5\right)\right)\right) \]
Alternative 6
Error0.7
Cost6720
\[2 \cdot \tan^{-1} \left(1 - x\right) \]
Alternative 7
Error1.4
Cost6592
\[2 \cdot \tan^{-1} 1 \]

Error

Reproduce?

herbie shell --seed 2023073 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))