Average Error: 0.0 → 0.0
Time: 6.9s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x}}{\sqrt{1 + x}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\frac{\sqrt{1 - x}}{\sqrt{1 + x}}\right)
double f(double x) {
        double r124853 = 2.0;
        double r124854 = 1.0;
        double r124855 = x;
        double r124856 = r124854 - r124855;
        double r124857 = r124854 + r124855;
        double r124858 = r124856 / r124857;
        double r124859 = sqrt(r124858);
        double r124860 = atan(r124859);
        double r124861 = r124853 * r124860;
        return r124861;
}
↓
double f(double x) {
        double r124862 = 2.0;
        double r124863 = 1.0;
        double r124864 = x;
        double r124865 = r124863 - r124864;
        double r124866 = sqrt(r124865);
        double r124867 = r124863 + r124864;
        double r124868 = sqrt(r124867);
        double r124869 = r124866 / r124868;
        double r124870 = atan(r124869);
        double r124871 = r124862 * r124870;
        return r124871;
}

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 sqrt-div0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))