Average Error: 0.0 → 0.0
Time: 14.0s
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 r586665 = 2.0;
        double r586666 = 1.0;
        double r586667 = x;
        double r586668 = r586666 - r586667;
        double r586669 = r586666 + r586667;
        double r586670 = r586668 / r586669;
        double r586671 = sqrt(r586670);
        double r586672 = atan(r586671);
        double r586673 = r586665 * r586672;
        return r586673;
}
↓
double f(double x) {
        double r586674 = 2.0;
        double r586675 = 1.0;
        double r586676 = x;
        double r586677 = r586675 - r586676;
        double r586678 = sqrt(r586677);
        double r586679 = r586675 + r586676;
        double r586680 = sqrt(r586679);
        double r586681 = r586678 / r586680;
        double r586682 = atan(r586681);
        double r586683 = r586674 * r586682;
        return r586683;
}

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 2019172 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))