Average Error: 0.0 → 0.0
Time: 28.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \cdot 2
double f(double x) {
        double r963087 = 2.0;
        double r963088 = 1.0;
        double r963089 = x;
        double r963090 = r963088 - r963089;
        double r963091 = r963088 + r963089;
        double r963092 = r963090 / r963091;
        double r963093 = sqrt(r963092);
        double r963094 = atan(r963093);
        double r963095 = r963087 * r963094;
        return r963095;
}
↓
double f(double x) {
        double r963096 = 1.0;
        double r963097 = x;
        double r963098 = r963096 - r963097;
        double r963099 = r963096 + r963097;
        double r963100 = r963098 / r963099;
        double r963101 = sqrt(r963100);
        double r963102 = atan(r963101);
        double r963103 = 2.0;
        double r963104 = r963102 * r963103;
        return r963104;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))