Average Error: 0.0 → 0.0
Time: 23.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{1 - x}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}}{\sqrt[3]{1 + x}}}\right)
double f(double x) {
        double r450088 = 2.0;
        double r450089 = 1.0;
        double r450090 = x;
        double r450091 = r450089 - r450090;
        double r450092 = r450089 + r450090;
        double r450093 = r450091 / r450092;
        double r450094 = sqrt(r450093);
        double r450095 = atan(r450094);
        double r450096 = r450088 * r450095;
        return r450096;
}
↓
double f(double x) {
        double r450097 = 2.0;
        double r450098 = 1.0;
        double r450099 = x;
        double r450100 = r450098 - r450099;
        double r450101 = r450098 + r450099;
        double r450102 = cbrt(r450101);
        double r450103 = r450102 * r450102;
        double r450104 = r450100 / r450103;
        double r450105 = r450104 / r450102;
        double r450106 = sqrt(r450105);
        double r450107 = atan(r450106);
        double r450108 = r450097 * r450107;
        return r450108;
}

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-cube-cbrt0.0

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

  4. Applied associate-/r*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))