Average Error: 0.0 → 0.0
Time: 11.9s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\left(1 - x\right) \cdot \frac{1}{1 + x}\right)}^{3}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\left(1 - x\right) \cdot \frac{1}{1 + x}\right)}^{3}}}\right)
double f(double x) {
        double r18623 = 2.0;
        double r18624 = 1.0;
        double r18625 = x;
        double r18626 = r18624 - r18625;
        double r18627 = r18624 + r18625;
        double r18628 = r18626 / r18627;
        double r18629 = sqrt(r18628);
        double r18630 = atan(r18629);
        double r18631 = r18623 * r18630;
        return r18631;
}
↓
double f(double x) {
        double r18632 = 2.0;
        double r18633 = 1.0;
        double r18634 = x;
        double r18635 = r18633 - r18634;
        double r18636 = 1.0;
        double r18637 = r18633 + r18634;
        double r18638 = r18636 / r18637;
        double r18639 = r18635 * r18638;
        double r18640 = 3.0;
        double r18641 = pow(r18639, r18640);
        double r18642 = cbrt(r18641);
        double r18643 = sqrt(r18642);
        double r18644 = atan(r18643);
        double r18645 = r18632 * r18644;
        return r18645;
}

Error

Bits error versus x

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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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied cbrt-undiv0.0

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

  6. Simplified0.0

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

  8. Applied div-inv0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))