Average Error: 0.0 → 0.0
Time: 2.2s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{e^{\mathsf{log1p}\left(-x\right) - \mathsf{log1p}\left(x\right)}}\right) \]
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (* 2.0 (atan (sqrt (exp (- (log1p (- x)) (log1p x)))))))
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(-x) - log1p(x)))));
}

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(-x) - Math.log1p(x)))));
}

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(-x) - math.log1p(x)))))

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(exp(Float64(log1p(Float64(-x)) - log1p(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_] := N[(2.0 * N[ArcTan[N[Sqrt[N[Exp[N[(N[Log[1 + (-x)], $MachinePrecision] - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $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(-x\right) - \mathsf{log1p}\left(x\right)}}\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 add-exp-log_binary640.0

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

  3. Applied add-exp-log_binary640.0

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

  4. Applied div-exp_binary640.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2022129 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))