Average Error: 0.0 → 0.0
Time: 25.3s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\left(\left(\sqrt[3]{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}} \cdot \sqrt[3]{\frac{1 - x}{1 + x}}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{1 - x}}{1 + x}}\right) \cdot \sqrt[3]{\frac{1 - x}{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\left(\left(\sqrt[3]{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}} \cdot \sqrt[3]{\frac{1 - x}{1 + x}}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{1 - x}}{1 + x}}\right) \cdot \sqrt[3]{\frac{1 - x}{1 + x}}}\right)
double f(double x) {
        double r518474 = 2.0;
        double r518475 = 1.0;
        double r518476 = x;
        double r518477 = r518475 - r518476;
        double r518478 = r518475 + r518476;
        double r518479 = r518477 / r518478;
        double r518480 = sqrt(r518479);
        double r518481 = atan(r518480);
        double r518482 = r518474 * r518481;
        return r518482;
}
↓
double f(double x) {
        double r518483 = 2.0;
        double r518484 = 1.0;
        double r518485 = x;
        double r518486 = r518484 - r518485;
        double r518487 = cbrt(r518486);
        double r518488 = r518487 * r518487;
        double r518489 = cbrt(r518488);
        double r518490 = r518484 + r518485;
        double r518491 = r518486 / r518490;
        double r518492 = cbrt(r518491);
        double r518493 = r518489 * r518492;
        double r518494 = r518487 / r518490;
        double r518495 = cbrt(r518494);
        double r518496 = r518493 * r518495;
        double r518497 = r518496 * r518492;
        double r518498 = sqrt(r518497);
        double r518499 = atan(r518498);
        double r518500 = r518483 * r518499;
        return r518500;
}

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

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

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

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied times-frac0.0

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

  8. Applied cbrt-prod0.0

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

  9. Applied associate-*r*0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))