Average Error: 0.0 → 0.0
Time: 13.9s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{{1}^{3} - {x}^{3}}{x + 1} \cdot \frac{1}{x \cdot \left(x + 1\right) + 1 \cdot 1}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{{1}^{3} - {x}^{3}}{x + 1} \cdot \frac{1}{x \cdot \left(x + 1\right) + 1 \cdot 1}}\right)
double f(double x) {
        double r20642 = 2.0;
        double r20643 = 1.0;
        double r20644 = x;
        double r20645 = r20643 - r20644;
        double r20646 = r20643 + r20644;
        double r20647 = r20645 / r20646;
        double r20648 = sqrt(r20647);
        double r20649 = atan(r20648);
        double r20650 = r20642 * r20649;
        return r20650;
}
↓
double f(double x) {
        double r20651 = 2.0;
        double r20652 = 1.0;
        double r20653 = 3.0;
        double r20654 = pow(r20652, r20653);
        double r20655 = x;
        double r20656 = pow(r20655, r20653);
        double r20657 = r20654 - r20656;
        double r20658 = r20655 + r20652;
        double r20659 = r20657 / r20658;
        double r20660 = 1.0;
        double r20661 = r20655 * r20658;
        double r20662 = r20652 * r20652;
        double r20663 = r20661 + r20662;
        double r20664 = r20660 / r20663;
        double r20665 = r20659 * r20664;
        double r20666 = sqrt(r20665);
        double r20667 = atan(r20666);
        double r20668 = r20651 * r20667;
        return r20668;
}

Error

Bits error versus x

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Results

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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

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

  6. Applied associate-/r/0.0

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

  7. Applied add-cube-cbrt0.0

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

  8. Applied times-frac0.0

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))