Average Error: 0.0 → 0.0
Time: 3.7s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\left|\sqrt[3]{\frac{1 - x}{1 + x}}\right| \cdot \sqrt{\sqrt[3]{\frac{1 - x}{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\left|\sqrt[3]{\frac{1 - x}{1 + x}}\right| \cdot \sqrt{\sqrt[3]{\frac{1 - x}{1 + x}}}\right)
double f(double x) {
        double r12094 = 2.0;
        double r12095 = 1.0;
        double r12096 = x;
        double r12097 = r12095 - r12096;
        double r12098 = r12095 + r12096;
        double r12099 = r12097 / r12098;
        double r12100 = sqrt(r12099);
        double r12101 = atan(r12100);
        double r12102 = r12094 * r12101;
        return r12102;
}
↓
double f(double x) {
        double r12103 = 2.0;
        double r12104 = 1.0;
        double r12105 = x;
        double r12106 = r12104 - r12105;
        double r12107 = r12104 + r12105;
        double r12108 = r12106 / r12107;
        double r12109 = cbrt(r12108);
        double r12110 = fabs(r12109);
        double r12111 = sqrt(r12109);
        double r12112 = r12110 * r12111;
        double r12113 = atan(r12112);
        double r12114 = r12103 * r12113;
        return r12114;
}

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-cube-cbrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))