Average Error: 0.0 → 0.0
Time: 11.9s
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 r21243 = 2.0;
        double r21244 = 1.0;
        double r21245 = x;
        double r21246 = r21244 - r21245;
        double r21247 = r21244 + r21245;
        double r21248 = r21246 / r21247;
        double r21249 = sqrt(r21248);
        double r21250 = atan(r21249);
        double r21251 = r21243 * r21250;
        return r21251;
}
↓
double f(double x) {
        double r21252 = 2.0;
        double r21253 = 1.0;
        double r21254 = 1.0;
        double r21255 = x;
        double r21256 = r21254 + r21255;
        double r21257 = r21254 - r21255;
        double r21258 = r21256 / r21257;
        double r21259 = r21253 / r21258;
        double r21260 = sqrt(r21259);
        double r21261 = atan(r21260);
        double r21262 = r21252 * r21261;
        return r21262;
}

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 clear-num0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))