Average Error: 0.0 → 0.0
Time: 10.4s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1}{1 + x}\right)}^{3}} - \frac{x}{1 + x}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1}{1 + x}\right)}^{3}} - \frac{x}{1 + x}}\right)
double f(double x) {
        double r19520 = 2.0;
        double r19521 = 1.0;
        double r19522 = x;
        double r19523 = r19521 - r19522;
        double r19524 = r19521 + r19522;
        double r19525 = r19523 / r19524;
        double r19526 = sqrt(r19525);
        double r19527 = atan(r19526);
        double r19528 = r19520 * r19527;
        return r19528;
}
↓
double f(double x) {
        double r19529 = 2.0;
        double r19530 = 1.0;
        double r19531 = x;
        double r19532 = r19530 + r19531;
        double r19533 = r19530 / r19532;
        double r19534 = 3.0;
        double r19535 = pow(r19533, r19534);
        double r19536 = cbrt(r19535);
        double r19537 = r19531 / r19532;
        double r19538 = r19536 - r19537;
        double r19539 = sqrt(r19538);
        double r19540 = atan(r19539);
        double r19541 = r19529 * r19540;
        return r19541;
}

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 div-sub0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied cbrt-undiv0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019212 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))