Average Error: 0.0 → 0.0
Time: 14.4s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}}}\right)
double f(double x) {
        double r487377 = 2.0;
        double r487378 = 1.0;
        double r487379 = x;
        double r487380 = r487378 - r487379;
        double r487381 = r487378 + r487379;
        double r487382 = r487380 / r487381;
        double r487383 = sqrt(r487382);
        double r487384 = atan(r487383);
        double r487385 = r487377 * r487384;
        return r487385;
}
↓
double f(double x) {
        double r487386 = 2.0;
        double r487387 = 1.0;
        double r487388 = x;
        double r487389 = r487387 - r487388;
        double r487390 = r487387 + r487388;
        double r487391 = cbrt(r487390);
        double r487392 = r487391 * r487391;
        double r487393 = r487389 / r487392;
        double r487394 = r487393 / r487391;
        double r487395 = sqrt(r487394);
        double r487396 = atan(r487395);
        double r487397 = r487386 * r487396;
        return r487397;
}

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

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

  4. Applied associate-/r*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))