Average Error: 0.0 → 0.0
Time: 7.7s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{1 - x}{\sqrt[3]{x + 1}}} \cdot \sqrt{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{1 - x}{\sqrt[3]{x + 1}}} \cdot \sqrt{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}\right) \cdot 2
double f(double x) {
        double r592628 = 2.0;
        double r592629 = 1.0;
        double r592630 = x;
        double r592631 = r592629 - r592630;
        double r592632 = r592629 + r592630;
        double r592633 = r592631 / r592632;
        double r592634 = sqrt(r592633);
        double r592635 = atan(r592634);
        double r592636 = r592628 * r592635;
        return r592636;
}
↓
double f(double x) {
        double r592637 = 1.0;
        double r592638 = x;
        double r592639 = r592637 - r592638;
        double r592640 = r592638 + r592637;
        double r592641 = cbrt(r592640);
        double r592642 = r592639 / r592641;
        double r592643 = sqrt(r592642);
        double r592644 = 1.0;
        double r592645 = r592641 * r592641;
        double r592646 = r592644 / r592645;
        double r592647 = sqrt(r592646);
        double r592648 = r592643 * r592647;
        double r592649 = atan(r592648);
        double r592650 = 2.0;
        double r592651 = r592649 * r592650;
        return r592651;
}

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

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

  5. Applied times-frac0.0

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

  6. Applied sqrt-prod0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))