Average Error: 0.0 → 0.0
Time: 4.8s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{{\left(\frac{1}{x + 1}\right)}^{3} - {\left(\frac{x}{x + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{1}{x + 1} + \frac{x}{x + 1}, \frac{1}{x + 1} \cdot \frac{1}{x + 1}\right)}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{{\left(\frac{1}{x + 1}\right)}^{3} - {\left(\frac{x}{x + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{1}{x + 1} + \frac{x}{x + 1}, \frac{1}{x + 1} \cdot \frac{1}{x + 1}\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(((pow((1.0 / (x + 1.0)), 3.0) - pow((x / (x + 1.0)), 3.0)) / fma((x / (x + 1.0)), ((1.0 / (x + 1.0)) + (x / (x + 1.0))), ((1.0 / (x + 1.0)) * (1.0 / (x + 1.0))))))));
}

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 div-sub0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  7. Applied flip3--0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{{\left(\frac{1}{x + 1}\right)}^{3} - {\left(\frac{x}{x + 1}\right)}^{3}}{\frac{1}{x + 1} \cdot \frac{1}{x + 1} + \left(\frac{x}{x + 1} \cdot \frac{x}{x + 1} + \frac{1}{x + 1} \cdot \frac{x}{x + 1}\right)}}}\right)\]

  8. Simplified0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{{\left(\frac{1}{x + 1}\right)}^{3} - {\left(\frac{x}{x + 1}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{1}{x + 1} + \frac{x}{x + 1}, \frac{1}{x + 1} \cdot \frac{1}{x + 1}\right)}}}\right)\]

  9. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{{\left(\frac{1}{x + 1}\right)}^{3} - {\left(\frac{x}{x + 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x}{x + 1}, \frac{1}{x + 1} + \frac{x}{x + 1}, \frac{1}{x + 1} \cdot \frac{1}{x + 1}\right)}}\right)\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))