Average Error: 0.0 → 0.0
Time: 22.2s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{1}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \frac{1 - x}{\sqrt[3]{1 + x}}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{1}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \frac{1 - x}{\sqrt[3]{1 + x}}}\right) \cdot 2
double f(double x) {
        double r786004 = 2.0;
        double r786005 = 1.0;
        double r786006 = x;
        double r786007 = r786005 - r786006;
        double r786008 = r786005 + r786006;
        double r786009 = r786007 / r786008;
        double r786010 = sqrt(r786009);
        double r786011 = atan(r786010);
        double r786012 = r786004 * r786011;
        return r786012;
}
↓
double f(double x) {
        double r786013 = 1.0;
        double r786014 = x;
        double r786015 = r786013 + r786014;
        double r786016 = cbrt(r786015);
        double r786017 = r786016 * r786016;
        double r786018 = r786013 / r786017;
        double r786019 = r786013 - r786014;
        double r786020 = r786019 / r786016;
        double r786021 = r786018 * r786020;
        double r786022 = sqrt(r786021);
        double r786023 = atan(r786022);
        double r786024 = 2.0;
        double r786025 = r786023 * r786024;
        return r786025;
}

Error

Bits error versus x

Try it out

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 *-un-lft-identity0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied distribute-lft-out--0.0

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

  7. Applied times-frac0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))