Average Error: 0.0 → 0.0
Time: 12.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}}{\frac{1 + x}{\sqrt[3]{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}} \cdot \sqrt[3]{\sqrt[3]{1 - x}}}}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}}{\frac{1 + x}{\sqrt[3]{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}} \cdot \sqrt[3]{\sqrt[3]{1 - x}}}}}\right) \cdot 2
double f(double x) {
        double r943737 = 2.0;
        double r943738 = 1.0;
        double r943739 = x;
        double r943740 = r943738 - r943739;
        double r943741 = r943738 + r943739;
        double r943742 = r943740 / r943741;
        double r943743 = sqrt(r943742);
        double r943744 = atan(r943743);
        double r943745 = r943737 * r943744;
        return r943745;
}
↓
double f(double x) {
        double r943746 = 1.0;
        double r943747 = x;
        double r943748 = r943746 - r943747;
        double r943749 = cbrt(r943748);
        double r943750 = r943749 * r943749;
        double r943751 = r943746 + r943747;
        double r943752 = cbrt(r943750);
        double r943753 = cbrt(r943749);
        double r943754 = r943752 * r943753;
        double r943755 = r943751 / r943754;
        double r943756 = r943750 / r943755;
        double r943757 = sqrt(r943756);
        double r943758 = atan(r943757);
        double r943759 = 2.0;
        double r943760 = r943758 * r943759;
        return r943760;
}

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

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

  4. Applied associate-/l*0.0

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied cbrt-prod0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))