Average Error: 0.0 → 0.0
Time: 12.2s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{1 + x} - \frac{x}{1 + x}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{1 + x} - \frac{x}{1 + x}}\right)
double f(double x) {
        double r285482 = 2.0;
        double r285483 = 1.0;
        double r285484 = x;
        double r285485 = r285483 - r285484;
        double r285486 = r285483 + r285484;
        double r285487 = r285485 / r285486;
        double r285488 = sqrt(r285487);
        double r285489 = atan(r285488);
        double r285490 = r285482 * r285489;
        return r285490;
}
↓
double f(double x) {
        double r285491 = 2.0;
        double r285492 = 1.0;
        double r285493 = x;
        double r285494 = r285492 + r285493;
        double r285495 = r285492 / r285494;
        double r285496 = r285493 / r285494;
        double r285497 = r285495 - r285496;
        double r285498 = sqrt(r285497);
        double r285499 = atan(r285498);
        double r285500 = r285491 * r285499;
        return r285500;
}

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 div-sub0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))