Average Error: 0.0 → 0.0
Time: 6.8s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{{\left(1 - x\right)}^{3}}{{\left(x + 1\right)}^{3}}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{{\left(1 - x\right)}^{3}}{{\left(x + 1\right)}^{3}}}}\right)
double f(double x) {
        double r11929 = 2.0;
        double r11930 = 1.0;
        double r11931 = x;
        double r11932 = r11930 - r11931;
        double r11933 = r11930 + r11931;
        double r11934 = r11932 / r11933;
        double r11935 = sqrt(r11934);
        double r11936 = atan(r11935);
        double r11937 = r11929 * r11936;
        return r11937;
}
↓
double f(double x) {
        double r11938 = 2.0;
        double r11939 = 1.0;
        double r11940 = x;
        double r11941 = r11939 - r11940;
        double r11942 = 3.0;
        double r11943 = pow(r11941, r11942);
        double r11944 = r11940 + r11939;
        double r11945 = pow(r11944, r11942);
        double r11946 = r11943 / r11945;
        double r11947 = cbrt(r11946);
        double r11948 = sqrt(r11947);
        double r11949 = atan(r11948);
        double r11950 = r11938 * r11949;
        return r11950;
}

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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied cbrt-undiv0.0

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

  6. Simplified0.0

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

  8. Applied cube-div0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))