Average Error: 0.0 → 0.0
Time: 9.5s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \frac{\sqrt{1 - x}}{\sqrt[3]{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \frac{\sqrt{1 - x}}{\sqrt[3]{1 + x}}}\right)
double f(double x) {
        double r22787 = 2.0;
        double r22788 = 1.0;
        double r22789 = x;
        double r22790 = r22788 - r22789;
        double r22791 = r22788 + r22789;
        double r22792 = r22790 / r22791;
        double r22793 = sqrt(r22792);
        double r22794 = atan(r22793);
        double r22795 = r22787 * r22794;
        return r22795;
}
↓
double f(double x) {
        double r22796 = 2.0;
        double r22797 = 1.0;
        double r22798 = x;
        double r22799 = r22797 - r22798;
        double r22800 = sqrt(r22799);
        double r22801 = r22797 + r22798;
        double r22802 = cbrt(r22801);
        double r22803 = r22802 * r22802;
        double r22804 = r22800 / r22803;
        double r22805 = r22800 / r22802;
        double r22806 = r22804 * r22805;
        double r22807 = sqrt(r22806);
        double r22808 = atan(r22807);
        double r22809 = r22796 * r22808;
        return r22809;
}

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 add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))