Average Error: 0.0 → 0.0
Time: 3.8s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - x}{1 + x}\right)\right)}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - x}{1 + x}\right)\right)}\right)
double code(double x) {
	return (2.0 * atan(sqrt(((1.0 - x) / (1.0 + x)))));
}
↓
double code(double x) {
	return (2.0 * atan(sqrt(log1p(expm1(((1.0 - x) / (1.0 + x)))))));
}

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. Using strategy rm

  3. Applied log1p-expm1-u0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - x}{1 + x}\right)\right)}}\right)\]

  4. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - x}{1 + x}\right)\right)}\right)\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))