Average Error: 0.0 → 0.0
Time: 4.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\left|\sqrt[3]{\frac{1 - x}{1 + x}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{1 - x}}{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\left|\sqrt[3]{\frac{1 - x}{1 + x}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{1 - x}}{1 + x}}}\right)
double f(double x) {
        double r16886 = 2.0;
        double r16887 = 1.0;
        double r16888 = x;
        double r16889 = r16887 - r16888;
        double r16890 = r16887 + r16888;
        double r16891 = r16889 / r16890;
        double r16892 = sqrt(r16891);
        double r16893 = atan(r16892);
        double r16894 = r16886 * r16893;
        return r16894;
}
↓
double f(double x) {
        double r16895 = 2.0;
        double r16896 = 1.0;
        double r16897 = x;
        double r16898 = r16896 - r16897;
        double r16899 = r16896 + r16897;
        double r16900 = r16898 / r16899;
        double r16901 = cbrt(r16900);
        double r16902 = fabs(r16901);
        double r16903 = cbrt(r16898);
        double r16904 = r16903 * r16903;
        double r16905 = 1.0;
        double r16906 = r16904 / r16905;
        double r16907 = cbrt(r16906);
        double r16908 = r16903 / r16899;
        double r16909 = cbrt(r16908);
        double r16910 = r16907 * r16909;
        double r16911 = sqrt(r16910);
        double r16912 = r16902 * r16911;
        double r16913 = atan(r16912);
        double r16914 = r16895 * r16913;
        return r16914;
}

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 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. Applied sqrt-prod0.0

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

  5. Simplified0.0

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

  7. Applied *-un-lft-identity0.0

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

  8. Applied add-cube-cbrt0.0

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

  9. Applied times-frac0.0

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

  10. Applied cbrt-prod0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020039 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))