Average Error: 0.0 → 0.0
Time: 7.5s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{1 + x} \cdot \left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right)}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{1 + x} \cdot \left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right)}}\right)
double f(double x) {
        double r458593 = 2.0;
        double r458594 = 1.0;
        double r458595 = x;
        double r458596 = r458594 - r458595;
        double r458597 = r458594 + r458595;
        double r458598 = r458596 / r458597;
        double r458599 = sqrt(r458598);
        double r458600 = atan(r458599);
        double r458601 = r458593 * r458600;
        return r458601;
}
↓
double f(double x) {
        double r458602 = 2.0;
        double r458603 = 1.0;
        double r458604 = x;
        double r458605 = r458603 - r458604;
        double r458606 = r458603 + r458604;
        double r458607 = r458605 / r458606;
        double r458608 = r458607 * r458607;
        double r458609 = r458607 * r458608;
        double r458610 = cbrt(r458609);
        double r458611 = sqrt(r458610);
        double r458612 = atan(r458611);
        double r458613 = r458602 * r458612;
        return r458613;
}

Error

Bits error versus x

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Your Program's Arguments

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 add-cbrt-cube0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\sqrt[3]{\left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right) \cdot \frac{1 - x}{1 + x}}}}\right)\]

  4. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{1 + x} \cdot \left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right)}}\right)\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))