Average Error: 0.0 → 0.0
Time: 12.0s
Precision: 64
\[2.0 \cdot \tan^{-1} \left(\sqrt{\frac{1.0 - x}{1.0 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{1.0 - x}{1.0 + x}}\right) \cdot 2.0\]
2.0 \cdot \tan^{-1} \left(\sqrt{\frac{1.0 - x}{1.0 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{1.0 - x}{1.0 + x}}\right) \cdot 2.0
double f(double x) {
        double r167453 = 2.0;
        double r167454 = 1.0;
        double r167455 = x;
        double r167456 = r167454 - r167455;
        double r167457 = r167454 + r167455;
        double r167458 = r167456 / r167457;
        double r167459 = sqrt(r167458);
        double r167460 = atan(r167459);
        double r167461 = r167453 * r167460;
        return r167461;
}
↓
double f(double x) {
        double r167462 = 1.0;
        double r167463 = x;
        double r167464 = r167462 - r167463;
        double r167465 = r167462 + r167463;
        double r167466 = r167464 / r167465;
        double r167467 = sqrt(r167466);
        double r167468 = atan(r167467);
        double r167469 = 2.0;
        double r167470 = r167468 * r167469;
        return r167470;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 0.0

    \[2.0 \cdot \tan^{-1} \left(\sqrt{\frac{1.0 - x}{1.0 + x}}\right)\]

  2. Final simplification0.0

    \[\leadsto \tan^{-1} \left(\sqrt{\frac{1.0 - x}{1.0 + x}}\right) \cdot 2.0\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))