Average Error: 0.0 → 0.0
Time: 23.5s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\left(\sqrt[3]{\frac{1 - x}{1 + x}} \cdot \sqrt[3]{\frac{1 - x}{1 + x}}\right) \cdot \sqrt[3]{\frac{1 - x}{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\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)
double f(double x) {
        double r825392 = 2.0;
        double r825393 = 1.0;
        double r825394 = x;
        double r825395 = r825393 - r825394;
        double r825396 = r825393 + r825394;
        double r825397 = r825395 / r825396;
        double r825398 = sqrt(r825397);
        double r825399 = atan(r825398);
        double r825400 = r825392 * r825399;
        return r825400;
}
↓
double f(double x) {
        double r825401 = 2.0;
        double r825402 = 1.0;
        double r825403 = x;
        double r825404 = r825402 - r825403;
        double r825405 = r825402 + r825403;
        double r825406 = r825404 / r825405;
        double r825407 = cbrt(r825406);
        double r825408 = r825407 * r825407;
        double r825409 = r825408 * r825407;
        double r825410 = sqrt(r825409);
        double r825411 = atan(r825410);
        double r825412 = r825401 * r825411;
        return r825412;
}

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-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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))