Average Error: 0.0 → 0.0
Time: 11.9s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt{1 + x}}}{\sqrt{1 + x}}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt{1 + x}}}{\sqrt{1 + x}}}\right) \cdot 2
double f(double x) {
        double r802214 = 2.0;
        double r802215 = 1.0;
        double r802216 = x;
        double r802217 = r802215 - r802216;
        double r802218 = r802215 + r802216;
        double r802219 = r802217 / r802218;
        double r802220 = sqrt(r802219);
        double r802221 = atan(r802220);
        double r802222 = r802214 * r802221;
        return r802222;
}
↓
double f(double x) {
        double r802223 = 1.0;
        double r802224 = x;
        double r802225 = r802223 - r802224;
        double r802226 = r802223 + r802224;
        double r802227 = sqrt(r802226);
        double r802228 = r802225 / r802227;
        double r802229 = r802228 / r802227;
        double r802230 = sqrt(r802229);
        double r802231 = atan(r802230);
        double r802232 = 2.0;
        double r802233 = r802231 * r802232;
        return r802233;
}

Error

Bits error versus x

Try it out

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 associate-/r*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))