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 r583115 = 2.0;
        double r583116 = 1.0;
        double r583117 = x;
        double r583118 = r583116 - r583117;
        double r583119 = r583116 + r583117;
        double r583120 = r583118 / r583119;
        double r583121 = sqrt(r583120);
        double r583122 = atan(r583121);
        double r583123 = r583115 * r583122;
        return r583123;
}
↓
double f(double x) {
        double r583124 = 2.0;
        double r583125 = 1.0;
        double r583126 = x;
        double r583127 = r583125 + r583126;
        double r583128 = r583125 / r583127;
        double r583129 = r583126 / r583127;
        double r583130 = r583128 - r583129;
        double r583131 = sqrt(r583130);
        double r583132 = atan(r583131);
        double r583133 = r583124 * r583132;
        return r583133;
}

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