Average Error: 0.0 → 0.0
Time: 22.1s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\left(\sqrt[3]{\frac{1 - x}{1 + x}} \cdot \sqrt[3]{\frac{1 - x}{1 + x}}\right) \cdot \sqrt[3]{\frac{1 - x}{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\left(\sqrt[3]{\frac{1 - x}{1 + x}} \cdot \sqrt[3]{\frac{1 - x}{1 + x}}\right) \cdot \sqrt[3]{\frac{1 - x}{1 + x}}}\right)
double f(double x) {
        double r916902 = 2.0;
        double r916903 = 1.0;
        double r916904 = x;
        double r916905 = r916903 - r916904;
        double r916906 = r916903 + r916904;
        double r916907 = r916905 / r916906;
        double r916908 = sqrt(r916907);
        double r916909 = atan(r916908);
        double r916910 = r916902 * r916909;
        return r916910;
}
↓
double f(double x) {
        double r916911 = 2.0;
        double r916912 = 1.0;
        double r916913 = x;
        double r916914 = r916912 - r916913;
        double r916915 = r916912 + r916913;
        double r916916 = r916914 / r916915;
        double r916917 = cbrt(r916916);
        double r916918 = r916917 * r916917;
        double r916919 = r916918 * r916917;
        double r916920 = sqrt(r916919);
        double r916921 = atan(r916920);
        double r916922 = r916911 * r916921;
        return r916922;
}

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-cube-cbrt0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019129 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))