Average Error: 0.0 → 0.0
Time: 8.1s
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 r172738 = 2.0;
        double r172739 = 1.0;
        double r172740 = x;
        double r172741 = r172739 - r172740;
        double r172742 = r172739 + r172740;
        double r172743 = r172741 / r172742;
        double r172744 = sqrt(r172743);
        double r172745 = atan(r172744);
        double r172746 = r172738 * r172745;
        return r172746;
}
↓
double f(double x) {
        double r172747 = 1.0;
        double r172748 = x;
        double r172749 = r172747 - r172748;
        double r172750 = r172747 + r172748;
        double r172751 = r172749 / r172750;
        double r172752 = sqrt(r172751);
        double r172753 = atan(r172752);
        double r172754 = 2.0;
        double r172755 = r172753 * r172754;
        return r172755;
}

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 2019154 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))