Average Error: 0.0 → 0.0
Time: 23.9s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \cdot 2
double f(double x) {
        double r291628 = 2.0;
        double r291629 = 1.0;
        double r291630 = x;
        double r291631 = r291629 - r291630;
        double r291632 = r291629 + r291630;
        double r291633 = r291631 / r291632;
        double r291634 = sqrt(r291633);
        double r291635 = atan(r291634);
        double r291636 = r291628 * r291635;
        return r291636;
}
↓
double f(double x) {
        double r291637 = 1.0;
        double r291638 = x;
        double r291639 = r291637 - r291638;
        double r291640 = r291637 + r291638;
        double r291641 = r291639 / r291640;
        double r291642 = sqrt(r291641);
        double r291643 = atan(r291642);
        double r291644 = 2.0;
        double r291645 = r291643 * r291644;
        return r291645;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 0.0

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

  2. Final simplification0.0

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))