Average Error: 0.0 → 0.0
Time: 11.8s
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 r583075 = 2.0;
        double r583076 = 1.0;
        double r583077 = x;
        double r583078 = r583076 - r583077;
        double r583079 = r583076 + r583077;
        double r583080 = r583078 / r583079;
        double r583081 = sqrt(r583080);
        double r583082 = atan(r583081);
        double r583083 = r583075 * r583082;
        return r583083;
}
↓
double f(double x) {
        double r583084 = 2.0;
        double r583085 = 1.0;
        double r583086 = x;
        double r583087 = r583085 - r583086;
        double r583088 = sqrt(r583087);
        double r583089 = r583085 + r583086;
        double r583090 = sqrt(r583089);
        double r583091 = r583088 / r583090;
        double r583092 = atan(r583091);
        double r583093 = r583084 * r583092;
        return r583093;
}

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