Average Error: 0.0 → 0.0
Time: 1.2m
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{1 + x} \cdot \left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right)}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{1 + x} \cdot \left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right)}}\right)
double f(double x) {
        double r2431022 = 2.0;
        double r2431023 = 1.0;
        double r2431024 = x;
        double r2431025 = r2431023 - r2431024;
        double r2431026 = r2431023 + r2431024;
        double r2431027 = r2431025 / r2431026;
        double r2431028 = sqrt(r2431027);
        double r2431029 = atan(r2431028);
        double r2431030 = r2431022 * r2431029;
        return r2431030;
}
↓
double f(double x) {
        double r2431031 = 2.0;
        double r2431032 = 1.0;
        double r2431033 = x;
        double r2431034 = r2431032 - r2431033;
        double r2431035 = r2431032 + r2431033;
        double r2431036 = r2431034 / r2431035;
        double r2431037 = r2431036 * r2431036;
        double r2431038 = r2431036 * r2431037;
        double r2431039 = cbrt(r2431038);
        double r2431040 = sqrt(r2431039);
        double r2431041 = atan(r2431040);
        double r2431042 = r2431031 * r2431041;
        return r2431042;
}

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 *-un-lft-identity0.0

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

  4. Applied associate-/l*0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied add-cbrt-cube0.0

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

  8. Applied cbrt-undiv0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))