Average Error: 0.0 → 0.0
Time: 7.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{1 + x} \cdot \left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right)}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{1 + x} \cdot \left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right)}}\right)
double f(double x) {
        double r521770 = 2.0;
        double r521771 = 1.0;
        double r521772 = x;
        double r521773 = r521771 - r521772;
        double r521774 = r521771 + r521772;
        double r521775 = r521773 / r521774;
        double r521776 = sqrt(r521775);
        double r521777 = atan(r521776);
        double r521778 = r521770 * r521777;
        return r521778;
}
↓
double f(double x) {
        double r521779 = 2.0;
        double r521780 = 1.0;
        double r521781 = x;
        double r521782 = r521780 - r521781;
        double r521783 = r521780 + r521781;
        double r521784 = r521782 / r521783;
        double r521785 = r521784 * r521784;
        double r521786 = r521784 * r521785;
        double r521787 = cbrt(r521786);
        double r521788 = sqrt(r521787);
        double r521789 = atan(r521788);
        double r521790 = r521779 * r521789;
        return r521790;
}

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 add-cbrt-cube0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))