Average Error: 0.0 → 0.4
Time: 1.8s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{e^{x \cdot -2}}\right) \]
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (exp (* x -2.0))))))
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((x * -2.0))));
}

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
    code = 2.0d0 * atan(sqrt(exp((x * (-2.0d0)))))
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) {
	return 2.0 * Math.atan(Math.sqrt(Math.exp((x * -2.0))));
}

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((x * -2.0))))

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(exp(Float64(x * -2.0)))))
end

function tmp = code(x)
	tmp = 2.0 * atan(sqrt(((1.0 - x) / (1.0 + x))));
end
↓
function tmp = code(x)
	tmp = 2.0 * atan(sqrt(exp((x * -2.0))));
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[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{e^{x \cdot -2}}\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. Taylor expanded in x around 0 0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))