Average Error: 0.0 → 0.0
Time: 41.4s
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 r1244657 = 2.0;
        double r1244658 = 1.0;
        double r1244659 = x;
        double r1244660 = r1244658 - r1244659;
        double r1244661 = r1244658 + r1244659;
        double r1244662 = r1244660 / r1244661;
        double r1244663 = sqrt(r1244662);
        double r1244664 = atan(r1244663);
        double r1244665 = r1244657 * r1244664;
        return r1244665;
}
↓
double f(double x) {
        double r1244666 = 2.0;
        double r1244667 = 1.0;
        double r1244668 = x;
        double r1244669 = r1244667 - r1244668;
        double r1244670 = r1244667 + r1244668;
        double r1244671 = cbrt(r1244670);
        double r1244672 = r1244671 * r1244671;
        double r1244673 = r1244669 / r1244672;
        double r1244674 = r1244673 / r1244671;
        double r1244675 = sqrt(r1244674);
        double r1244676 = atan(r1244675);
        double r1244677 = r1244666 * r1244676;
        return r1244677;
}

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 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 2019125 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))