Average Error: 0.0 → 0.0
Time: 5.7s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt{1 + x}}}{\frac{\sqrt{1 \cdot 1 - x \cdot x}}{\sqrt{1 - x}}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt{1 + x}}}{\frac{\sqrt{1 \cdot 1 - x \cdot x}}{\sqrt{1 - x}}}}\right)
double f(double x) {
        double r9286 = 2.0;
        double r9287 = 1.0;
        double r9288 = x;
        double r9289 = r9287 - r9288;
        double r9290 = r9287 + r9288;
        double r9291 = r9289 / r9290;
        double r9292 = sqrt(r9291);
        double r9293 = atan(r9292);
        double r9294 = r9286 * r9293;
        return r9294;
}
↓
double f(double x) {
        double r9295 = 2.0;
        double r9296 = 1.0;
        double r9297 = x;
        double r9298 = r9296 - r9297;
        double r9299 = r9296 + r9297;
        double r9300 = sqrt(r9299);
        double r9301 = r9298 / r9300;
        double r9302 = r9296 * r9296;
        double r9303 = r9297 * r9297;
        double r9304 = r9302 - r9303;
        double r9305 = sqrt(r9304);
        double r9306 = sqrt(r9298);
        double r9307 = r9305 / r9306;
        double r9308 = r9301 / r9307;
        double r9309 = sqrt(r9308);
        double r9310 = atan(r9309);
        double r9311 = r9295 * r9310;
        return r9311;
}

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. Using strategy rm

  6. Applied flip-+0.0

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

  7. Applied sqrt-div0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))