Average Error: 0.0 → 0.0
Time: 2.1s
Precision: binary64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{\left(1 - x\right) \cdot \left(\left(1 - x\right) \cdot \left(1 - x\right)\right)}{\left(1 + x\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}}}\right) \]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{\left(1 - x\right) \cdot \left(\left(1 - x\right) \cdot \left(1 - x\right)\right)}{\left(1 + x\right) \cdot \left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}}}\right)

(FPCore (x) :precision binary64 (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))
↓
(FPCore (x)
 :precision binary64
 (*
  2.0
  (atan
   (sqrt
    (cbrt
     (/
      (* (- 1.0 x) (* (- 1.0 x) (- 1.0 x)))
      (* (+ 1.0 x) (* (+ 1.0 x) (+ 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(cbrt(((1.0 - x) * ((1.0 - x) * (1.0 - x))) / ((1.0 + x) * ((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 add-cbrt-cube_binary640.0

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

  4. Applied add-cbrt-cube_binary640.0

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

  5. Applied cbrt-undiv_binary640.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2021205 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))