Average Error: 0.0 → 0.0
Time: 12.6s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}}{\frac{1 + x}{\sqrt[3]{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}} \cdot \sqrt[3]{\sqrt[3]{1 - x}}}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}}{\frac{1 + x}{\sqrt[3]{\sqrt[3]{1 - x} \cdot \sqrt[3]{1 - x}} \cdot \sqrt[3]{\sqrt[3]{1 - x}}}}}\right)
double f(double x) {
        double r956533 = 2.0;
        double r956534 = 1.0;
        double r956535 = x;
        double r956536 = r956534 - r956535;
        double r956537 = r956534 + r956535;
        double r956538 = r956536 / r956537;
        double r956539 = sqrt(r956538);
        double r956540 = atan(r956539);
        double r956541 = r956533 * r956540;
        return r956541;
}
↓
double f(double x) {
        double r956542 = 2.0;
        double r956543 = 1.0;
        double r956544 = x;
        double r956545 = r956543 - r956544;
        double r956546 = cbrt(r956545);
        double r956547 = r956546 * r956546;
        double r956548 = r956543 + r956544;
        double r956549 = cbrt(r956547);
        double r956550 = cbrt(r956546);
        double r956551 = r956549 * r956550;
        double r956552 = r956548 / r956551;
        double r956553 = r956547 / r956552;
        double r956554 = sqrt(r956553);
        double r956555 = atan(r956554);
        double r956556 = r956542 * r956555;
        return r956556;
}

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

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

  4. Applied associate-/l*0.0

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

  6. Applied add-cube-cbrt0.0

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

  7. Applied cbrt-prod0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))