Average Error: 0.0 → 0.0
Time: 8.1s
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 r17108 = 2.0;
        double r17109 = 1.0;
        double r17110 = x;
        double r17111 = r17109 - r17110;
        double r17112 = r17109 + r17110;
        double r17113 = r17111 / r17112;
        double r17114 = sqrt(r17113);
        double r17115 = atan(r17114);
        double r17116 = r17108 * r17115;
        return r17116;
}
↓
double f(double x) {
        double r17117 = 2.0;
        double r17118 = 1.0;
        double r17119 = x;
        double r17120 = r17118 - r17119;
        double r17121 = r17118 + r17119;
        double r17122 = cbrt(r17121);
        double r17123 = r17122 * r17122;
        double r17124 = r17120 / r17123;
        double r17125 = r17124 / r17122;
        double r17126 = sqrt(r17125);
        double r17127 = atan(r17126);
        double r17128 = r17117 * r17127;
        return r17128;
}

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 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 2019209 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))