Average Error: 0.0 → 0.0
Time: 5.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\frac{{1}^{3} + {x}^{3}}{\left(1 - x\right) \cdot \mathsf{fma}\left(x, x - 1, 1 \cdot 1\right)}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\frac{{1}^{3} + {x}^{3}}{\left(1 - x\right) \cdot \mathsf{fma}\left(x, x - 1, 1 \cdot 1\right)}}}\right)
double code(double x) {
	return ((double) (2.0 * ((double) atan(((double) sqrt(((double) (((double) (1.0 - x)) / ((double) (1.0 + x))))))))));
}
↓
double code(double x) {
	return ((double) (2.0 * ((double) atan(((double) sqrt(((double) (1.0 / ((double) (((double) (((double) pow(1.0, 3.0)) + ((double) pow(x, 3.0)))) / ((double) (((double) (1.0 - x)) * ((double) fma(x, ((double) (x - 1.0)), ((double) (1.0 * 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 clear-num0.0

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

  5. Applied flip3-+0.0

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

  6. Applied associate-/l/0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))