Average Error: 0.0 → 0.0
Time: 2.8m
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 r10822851 = 2.0;
        double r10822852 = 1.0;
        double r10822853 = x;
        double r10822854 = r10822852 - r10822853;
        double r10822855 = r10822852 + r10822853;
        double r10822856 = r10822854 / r10822855;
        double r10822857 = sqrt(r10822856);
        double r10822858 = atan(r10822857);
        double r10822859 = r10822851 * r10822858;
        return r10822859;
}
↓
double f(double x) {
        double r10822860 = 1.0;
        double r10822861 = x;
        double r10822862 = r10822860 - r10822861;
        double r10822863 = r10822861 * r10822861;
        double r10822864 = r10822860 - r10822863;
        double r10822865 = r10822862 / r10822864;
        double r10822866 = r10822862 * r10822865;
        double r10822867 = sqrt(r10822866);
        double r10822868 = r10822867 * r10822867;
        double r10822869 = r10822867 * r10822868;
        double r10822870 = cbrt(r10822869);
        double r10822871 = atan(r10822870);
        double r10822872 = 2.0;
        double r10822873 = r10822871 * r10822872;
        return r10822873;
}

Error

Bits error versus x

Try it out

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))))))