Average Error: 0.0 → 0.0
Time: 2.7m
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt[3]{\sqrt{\left(1 - x\right) \cdot \frac{1 - x}{1 - x \cdot x}} \cdot \left(\sqrt{\left(1 - x\right) \cdot \frac{1 - x}{1 - x \cdot x}} \cdot \sqrt{\left(1 - x\right) \cdot \frac{1 - x}{1 - x \cdot x}}\right)}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt[3]{\sqrt{\left(1 - x\right) \cdot \frac{1 - x}{1 - x \cdot x}} \cdot \left(\sqrt{\left(1 - x\right) \cdot \frac{1 - x}{1 - x \cdot x}} \cdot \sqrt{\left(1 - x\right) \cdot \frac{1 - x}{1 - x \cdot x}}\right)}\right) \cdot 2
double f(double x) {
        double r10822846 = 2.0;
        double r10822847 = 1.0;
        double r10822848 = x;
        double r10822849 = r10822847 - r10822848;
        double r10822850 = r10822847 + r10822848;
        double r10822851 = r10822849 / r10822850;
        double r10822852 = sqrt(r10822851);
        double r10822853 = atan(r10822852);
        double r10822854 = r10822846 * r10822853;
        return r10822854;
}
↓
double f(double x) {
        double r10822855 = 1.0;
        double r10822856 = x;
        double r10822857 = r10822855 - r10822856;
        double r10822858 = r10822856 * r10822856;
        double r10822859 = r10822855 - r10822858;
        double r10822860 = r10822857 / r10822859;
        double r10822861 = r10822857 * r10822860;
        double r10822862 = sqrt(r10822861);
        double r10822863 = r10822862 * r10822862;
        double r10822864 = r10822862 * r10822863;
        double r10822865 = cbrt(r10822864);
        double r10822866 = atan(r10822865);
        double r10822867 = 2.0;
        double r10822868 = r10822866 * r10822867;
        return r10822868;
}

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 flip-+0.0

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

  4. Applied associate-/r/0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019107 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))