Average Error: 0.0 → 0.0
Time: 43.4s
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 r1359693 = 2.0;
        double r1359694 = 1.0;
        double r1359695 = x;
        double r1359696 = r1359694 - r1359695;
        double r1359697 = r1359694 + r1359695;
        double r1359698 = r1359696 / r1359697;
        double r1359699 = sqrt(r1359698);
        double r1359700 = atan(r1359699);
        double r1359701 = r1359693 * r1359700;
        return r1359701;
}
↓
double f(double x) {
        double r1359702 = 2.0;
        double r1359703 = 1.0;
        double r1359704 = x;
        double r1359705 = r1359703 - r1359704;
        double r1359706 = r1359703 + r1359704;
        double r1359707 = cbrt(r1359706);
        double r1359708 = r1359707 * r1359707;
        double r1359709 = r1359705 / r1359708;
        double r1359710 = r1359709 / r1359707;
        double r1359711 = sqrt(r1359710);
        double r1359712 = atan(r1359711);
        double r1359713 = r1359702 * r1359712;
        return r1359713;
}

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 2019125 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))