Average Error: 0.0 → 0.0
Time: 12.7s
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 r608353 = 2.0;
        double r608354 = 1.0;
        double r608355 = x;
        double r608356 = r608354 - r608355;
        double r608357 = r608354 + r608355;
        double r608358 = r608356 / r608357;
        double r608359 = sqrt(r608358);
        double r608360 = atan(r608359);
        double r608361 = r608353 * r608360;
        return r608361;
}
↓
double f(double x) {
        double r608362 = 2.0;
        double r608363 = 1.0;
        double r608364 = x;
        double r608365 = r608363 - r608364;
        double r608366 = sqrt(r608365);
        double r608367 = r608363 + r608364;
        double r608368 = sqrt(r608367);
        double r608369 = r608366 / r608368;
        double r608370 = atan(r608369);
        double r608371 = r608362 * r608370;
        return r608371;
}

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 2019163 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))