Average Error: 0.0 → 0.0
Time: 1.3m
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + x}} \cdot \sqrt[3]{\frac{1 - x}{\sqrt{1 + x}} \cdot \left(\frac{1 - x}{\sqrt{1 + x}} \cdot \frac{1 - x}{\sqrt{1 + x}}\right)}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + x}} \cdot \sqrt[3]{\frac{1 - x}{\sqrt{1 + x}} \cdot \left(\frac{1 - x}{\sqrt{1 + x}} \cdot \frac{1 - x}{\sqrt{1 + x}}\right)}}\right) \cdot 2
double f(double x) {
        double r2756718 = 2.0;
        double r2756719 = 1.0;
        double r2756720 = x;
        double r2756721 = r2756719 - r2756720;
        double r2756722 = r2756719 + r2756720;
        double r2756723 = r2756721 / r2756722;
        double r2756724 = sqrt(r2756723);
        double r2756725 = atan(r2756724);
        double r2756726 = r2756718 * r2756725;
        return r2756726;
}
↓
double f(double x) {
        double r2756727 = 1.0;
        double r2756728 = x;
        double r2756729 = r2756727 + r2756728;
        double r2756730 = sqrt(r2756729);
        double r2756731 = r2756727 / r2756730;
        double r2756732 = r2756727 - r2756728;
        double r2756733 = r2756732 / r2756730;
        double r2756734 = r2756733 * r2756733;
        double r2756735 = r2756733 * r2756734;
        double r2756736 = cbrt(r2756735);
        double r2756737 = r2756731 * r2756736;
        double r2756738 = sqrt(r2756737);
        double r2756739 = atan(r2756738);
        double r2756740 = 2.0;
        double r2756741 = r2756739 * r2756740;
        return r2756741;
}

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-sqr-sqrt0.0

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

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

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

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

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

  6. Applied distribute-lft-out--0.0

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

  7. Applied times-frac0.0

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

  9. Applied add-cbrt-cube0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))