arccos

Average Error: 0.0 → 0.0

Time: 4.0s

Precision: 64

Math

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

↓

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

C

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)) / (((x / (x + 1.0)) * ((x / (x + 1.0)) + (1.0 / (x + 1.0)))) + ((1.0 / (x + 1.0)) * (1.0 / (x + 1.0))))))));
}

Error

Bits error versus x

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-sub 0.0

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

4. Simplified 0.0

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

5. Simplified 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{x + 1} - \color{blue}{\frac{x}{x + 1}}}\right)\]
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. Simplified 0.0

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

9. Final simplification 0.0

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

Reproduce

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