Average Error: 0.0 → 0.0
Time: 19.5s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{1 - x}{\left(x + 1\right) \cdot \left(1 - x\right)} \cdot \left(1 - x\right)}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{1 - x}{\left(x + 1\right) \cdot \left(1 - x\right)} \cdot \left(1 - x\right)}\right) \cdot 2
double f(double x) {
        double r787007 = 2.0;
        double r787008 = 1.0;
        double r787009 = x;
        double r787010 = r787008 - r787009;
        double r787011 = r787008 + r787009;
        double r787012 = r787010 / r787011;
        double r787013 = sqrt(r787012);
        double r787014 = atan(r787013);
        double r787015 = r787007 * r787014;
        return r787015;
}
↓
double f(double x) {
        double r787016 = 1.0;
        double r787017 = x;
        double r787018 = r787016 - r787017;
        double r787019 = r787017 + r787016;
        double r787020 = r787019 * r787018;
        double r787021 = r787018 / r787020;
        double r787022 = r787021 * r787018;
        double r787023 = sqrt(r787022);
        double r787024 = atan(r787023);
        double r787025 = 2.0;
        double r787026 = r787024 * r787025;
        return r787026;
}

Error

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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 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. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))