Average Error: 0.0 → 0.0
Time: 2.8s
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 r8795 = 2.0;
        double r8796 = 1.0;
        double r8797 = x;
        double r8798 = r8796 - r8797;
        double r8799 = r8796 + r8797;
        double r8800 = r8798 / r8799;
        double r8801 = sqrt(r8800);
        double r8802 = atan(r8801);
        double r8803 = r8795 * r8802;
        return r8803;
}
↓
double f(double x) {
        double r8804 = 2.0;
        double r8805 = 1.0;
        double r8806 = x;
        double r8807 = r8805 - r8806;
        double r8808 = r8805 + r8806;
        double r8809 = sqrt(r8808);
        double r8810 = r8807 / r8809;
        double r8811 = r8805 * r8805;
        double r8812 = r8806 * r8806;
        double r8813 = r8811 - r8812;
        double r8814 = sqrt(r8813);
        double r8815 = sqrt(r8807);
        double r8816 = r8814 / r8815;
        double r8817 = r8810 / r8816;
        double r8818 = sqrt(r8817);
        double r8819 = atan(r8818);
        double r8820 = r8804 * r8819;
        return r8820;
}

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-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))))))