Average Error: 0.0 → 0.0
Time: 4.7s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{{1}^{3} - {x}^{3}}{\left(x \cdot \left(x + 1\right) + 1 \cdot 1\right) \cdot \left(1 + x\right)}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{{1}^{3} - {x}^{3}}{\left(x \cdot \left(x + 1\right) + 1 \cdot 1\right) \cdot \left(1 + x\right)}}\right)
double f(double x) {
        double r16712 = 2.0;
        double r16713 = 1.0;
        double r16714 = x;
        double r16715 = r16713 - r16714;
        double r16716 = r16713 + r16714;
        double r16717 = r16715 / r16716;
        double r16718 = sqrt(r16717);
        double r16719 = atan(r16718);
        double r16720 = r16712 * r16719;
        return r16720;
}
↓
double f(double x) {
        double r16721 = 2.0;
        double r16722 = 1.0;
        double r16723 = 3.0;
        double r16724 = pow(r16722, r16723);
        double r16725 = x;
        double r16726 = pow(r16725, r16723);
        double r16727 = r16724 - r16726;
        double r16728 = r16725 + r16722;
        double r16729 = r16725 * r16728;
        double r16730 = r16722 * r16722;
        double r16731 = r16729 + r16730;
        double r16732 = r16722 + r16725;
        double r16733 = r16731 * r16732;
        double r16734 = r16727 / r16733;
        double r16735 = sqrt(r16734);
        double r16736 = atan(r16735);
        double r16737 = r16721 * r16736;
        return r16737;
}

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

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

  4. Applied associate-/l/0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020034 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))