Average Error: 0.0 → 0.0
Time: 4.4s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\left|\sqrt[3]{\frac{1 - x}{1 + x}}\right| \cdot \sqrt{\sqrt[3]{\frac{1 - x}{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\left|\sqrt[3]{\frac{1 - x}{1 + x}}\right| \cdot \sqrt{\sqrt[3]{\frac{1 - x}{1 + x}}}\right)
double f(double x) {
        double r14470 = 2.0;
        double r14471 = 1.0;
        double r14472 = x;
        double r14473 = r14471 - r14472;
        double r14474 = r14471 + r14472;
        double r14475 = r14473 / r14474;
        double r14476 = sqrt(r14475);
        double r14477 = atan(r14476);
        double r14478 = r14470 * r14477;
        return r14478;
}
↓
double f(double x) {
        double r14479 = 2.0;
        double r14480 = 1.0;
        double r14481 = x;
        double r14482 = r14480 - r14481;
        double r14483 = r14480 + r14481;
        double r14484 = r14482 / r14483;
        double r14485 = cbrt(r14484);
        double r14486 = fabs(r14485);
        double r14487 = sqrt(r14485);
        double r14488 = r14486 * r14487;
        double r14489 = atan(r14488);
        double r14490 = r14479 * r14489;
        return r14490;
}

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. Applied sqrt-prod0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))