Average Error: 0.0 → 0.0
Time: 11.1s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(e^{\log \left(\sqrt{\frac{1 - x}{1 + x}}\right)}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(e^{\log \left(\sqrt{\frac{1 - x}{1 + x}}\right)}\right) \cdot 2
double f(double x) {
        double r226717 = 2.0;
        double r226718 = 1.0;
        double r226719 = x;
        double r226720 = r226718 - r226719;
        double r226721 = r226718 + r226719;
        double r226722 = r226720 / r226721;
        double r226723 = sqrt(r226722);
        double r226724 = atan(r226723);
        double r226725 = r226717 * r226724;
        return r226725;
}
↓
double f(double x) {
        double r226726 = 1.0;
        double r226727 = x;
        double r226728 = r226726 - r226727;
        double r226729 = r226726 + r226727;
        double r226730 = r226728 / r226729;
        double r226731 = sqrt(r226730);
        double r226732 = log(r226731);
        double r226733 = exp(r226732);
        double r226734 = atan(r226733);
        double r226735 = 2.0;
        double r226736 = r226734 * r226735;
        return r226736;
}

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 add-exp-log0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019143 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))