Average Error: 0.0 → 0.0
Time: 10.6s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\left(1 - x\right) \cdot \frac{1}{x + 1}\right)}^{3}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\left(1 - x\right) \cdot \frac{1}{x + 1}\right)}^{3}}}\right)
double f(double x) {
        double r18834 = 2.0;
        double r18835 = 1.0;
        double r18836 = x;
        double r18837 = r18835 - r18836;
        double r18838 = r18835 + r18836;
        double r18839 = r18837 / r18838;
        double r18840 = sqrt(r18839);
        double r18841 = atan(r18840);
        double r18842 = r18834 * r18841;
        return r18842;
}
↓
double f(double x) {
        double r18843 = 2.0;
        double r18844 = 1.0;
        double r18845 = x;
        double r18846 = r18844 - r18845;
        double r18847 = 1.0;
        double r18848 = r18845 + r18844;
        double r18849 = r18847 / r18848;
        double r18850 = r18846 * r18849;
        double r18851 = 3.0;
        double r18852 = pow(r18850, r18851);
        double r18853 = cbrt(r18852);
        double r18854 = sqrt(r18853);
        double r18855 = atan(r18854);
        double r18856 = r18843 * r18855;
        return r18856;
}

Error

Bits error versus x

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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 add-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied cbrt-undiv0.0

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

  6. Simplified0.0

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

  8. Applied div-inv0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))