Average Error: 0.0 → 0.0
Time: 13.8s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - x}{1 + x}\right)}^{2}} \cdot \sqrt[3]{\frac{1 - x}{1 + x}}\right)}^{3}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1 - x}{1 + x}\right)}^{2}} \cdot \sqrt[3]{\frac{1 - x}{1 + x}}\right)}^{3}}}\right)
double f(double x) {
        double r25888 = 2.0;
        double r25889 = 1.0;
        double r25890 = x;
        double r25891 = r25889 - r25890;
        double r25892 = r25889 + r25890;
        double r25893 = r25891 / r25892;
        double r25894 = sqrt(r25893);
        double r25895 = atan(r25894);
        double r25896 = r25888 * r25895;
        return r25896;
}
↓
double f(double x) {
        double r25897 = 2.0;
        double r25898 = 1.0;
        double r25899 = x;
        double r25900 = r25898 - r25899;
        double r25901 = r25898 + r25899;
        double r25902 = r25900 / r25901;
        double r25903 = 2.0;
        double r25904 = pow(r25902, r25903);
        double r25905 = cbrt(r25904);
        double r25906 = cbrt(r25902);
        double r25907 = r25905 * r25906;
        double r25908 = 3.0;
        double r25909 = pow(r25907, r25908);
        double r25910 = cbrt(r25909);
        double r25911 = sqrt(r25910);
        double r25912 = atan(r25911);
        double r25913 = r25897 * r25912;
        return r25913;
}

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-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

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

  8. Applied add-cube-cbrt0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))