Average Error: 0.0 → 0.0
Time: 2.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\left|\frac{\sqrt{1 - x}}{\sqrt{1 + x}}\right|\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\left|\frac{\sqrt{1 - x}}{\sqrt{1 + x}}\right|\right)
double f(double x) {
        double r4893 = 2.0;
        double r4894 = 1.0;
        double r4895 = x;
        double r4896 = r4894 - r4895;
        double r4897 = r4894 + r4895;
        double r4898 = r4896 / r4897;
        double r4899 = sqrt(r4898);
        double r4900 = atan(r4899);
        double r4901 = r4893 * r4900;
        return r4901;
}
↓
double f(double x) {
        double r4902 = 2.0;
        double r4903 = 1.0;
        double r4904 = x;
        double r4905 = r4903 - r4904;
        double r4906 = sqrt(r4905);
        double r4907 = r4903 + r4904;
        double r4908 = sqrt(r4907);
        double r4909 = r4906 / r4908;
        double r4910 = fabs(r4909);
        double r4911 = atan(r4910);
        double r4912 = r4902 * r4911;
        return r4912;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sqr-sqrt0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied rem-sqrt-square0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))