Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{e^{\mathsf{log1p}\left(\frac{1 - x}{1 + x}\right)} + -1}\right) \]
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (* 2.0 (atan (sqrt (+ (exp (log1p (/ (- 1.0 x) (+ 1.0 x)))) -1.0)))))
double code(double x) {
	return 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
}
↓
double code(double x) {
	return 2.0 * atan(sqrt((exp(log1p(((1.0 - x) / (1.0 + x)))) + -1.0)));
}

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) {
	return 2.0 * Math.atan(Math.sqrt((Math.exp(Math.log1p(((1.0 - x) / (1.0 + x)))) + -1.0)));
}

def code(x):
	return 2.0 * math.atan(math.sqrt(((1.0 - x) / (1.0 + x))))
↓
def code(x):
	return 2.0 * math.atan(math.sqrt((math.exp(math.log1p(((1.0 - x) / (1.0 + x)))) + -1.0)))

function code(x)
	return Float64(2.0 * atan(sqrt(Float64(Float64(1.0 - x) / Float64(1.0 + x)))))
end
↓
function code(x)
	return Float64(2.0 * atan(sqrt(Float64(exp(log1p(Float64(Float64(1.0 - x) / Float64(1.0 + x)))) + -1.0))))
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_] := N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Exp[N[Log[1 + N[(N[(1.0 - x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{e^{\mathsf{log1p}\left(\frac{1 - x}{1 + x}\right)} + -1}\right)

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 egg-rr0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1 - x}{1 + x}\right)} - 1}}\right) \]

  3. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{e^{\mathsf{log1p}\left(\frac{1 - x}{1 + x}\right)} + -1}\right) \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))