Average Error: 0.0 → 0.0
Time: 4.8s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\frac{\sqrt{\left(1 \cdot 1 - x \cdot x\right) \cdot \left({\left(1 \cdot 1\right)}^{3} + {\left(x \cdot x - 1 \cdot x\right)}^{3}\right)}}{\sqrt{\left({1}^{3} + {x}^{3}\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(x \cdot x - 1 \cdot x\right) \cdot \left(x \cdot x - 1 \cdot x\right) - \left(1 \cdot 1\right) \cdot \left(x \cdot x - 1 \cdot x\right)\right)\right)} \cdot \sqrt{1 + x}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\frac{\sqrt{\left(1 \cdot 1 - x \cdot x\right) \cdot \left({\left(1 \cdot 1\right)}^{3} + {\left(x \cdot x - 1 \cdot x\right)}^{3}\right)}}{\sqrt{\left({1}^{3} + {x}^{3}\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(x \cdot x - 1 \cdot x\right) \cdot \left(x \cdot x - 1 \cdot x\right) - \left(1 \cdot 1\right) \cdot \left(x \cdot x - 1 \cdot x\right)\right)\right)} \cdot \sqrt{1 + x}}\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((((1.0 * 1.0) - (x * x)) * (pow((1.0 * 1.0), 3.0) + pow(((x * x) - (1.0 * x)), 3.0)))) / (sqrt(((pow(1.0, 3.0) + pow(x, 3.0)) * (((1.0 * 1.0) * (1.0 * 1.0)) + ((((x * x) - (1.0 * x)) * ((x * x) - (1.0 * x))) - ((1.0 * 1.0) * ((x * x) - (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 flip3-+0.0

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

  4. Applied associate-/r/0.0

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

  6. Applied flip3-+0.0

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

  7. Applied frac-times0.0

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

  8. Applied sqrt-div0.0

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

  10. Applied flip--0.0

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

  11. Applied associate-*l/0.0

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

  12. Applied sqrt-div0.0

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

  13. Applied associate-/l/0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))