Average Error: 0.0 → 0.0
Time: 15.2s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\frac{1 + x}{1 - x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\frac{1 + x}{1 - x}}}\right)
double f(double x) {
        double r536989 = 2.0;
        double r536990 = 1.0;
        double r536991 = x;
        double r536992 = r536990 - r536991;
        double r536993 = r536990 + r536991;
        double r536994 = r536992 / r536993;
        double r536995 = sqrt(r536994);
        double r536996 = atan(r536995);
        double r536997 = r536989 * r536996;
        return r536997;
}
↓
double f(double x) {
        double r536998 = 2.0;
        double r536999 = 1.0;
        double r537000 = x;
        double r537001 = r536999 + r537000;
        double r537002 = r536999 - r537000;
        double r537003 = r537001 / r537002;
        double r537004 = r536999 / r537003;
        double r537005 = sqrt(r537004);
        double r537006 = atan(r537005);
        double r537007 = r536998 * r537006;
        return r537007;
}

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. Using strategy rm

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

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

  4. Applied associate-/l*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))