Average Error: 0.0 → 0.0
Time: 5.7s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x \cdot x} \cdot \sqrt{\frac{1}{1 + x}}}{\sqrt{1 + x}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x \cdot x} \cdot \sqrt{\frac{1}{1 + x}}}{\sqrt{1 + x}}\right)
(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (*
  2.0
  (atan
   (/ (* (sqrt (- 1.0 (* x x))) (sqrt (/ 1.0 (+ 1.0 x)))) (sqrt (+ 1.0 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(1.0 - (x * x)) * sqrt(1.0 / (1.0 + x))) / sqrt(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 div-inv_binary640.0

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

  4. Applied sqrt-prod_binary640.0

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

  6. Applied flip--_binary640.0

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

  7. Applied sqrt-div_binary640.0

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

  8. Applied associate-*l/_binary640.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2021050 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))