Average Error: 0.0 → 0.0
Time: 5.0s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{{1}^{3} + {x}^{3}} \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{{1}^{3} + {x}^{3}} \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}\right)
double f(double x) {
        double r22131 = 2.0;
        double r22132 = 1.0;
        double r22133 = x;
        double r22134 = r22132 - r22133;
        double r22135 = r22132 + r22133;
        double r22136 = r22134 / r22135;
        double r22137 = sqrt(r22136);
        double r22138 = atan(r22137);
        double r22139 = r22131 * r22138;
        return r22139;
}
↓
double f(double x) {
        double r22140 = 2.0;
        double r22141 = 1.0;
        double r22142 = x;
        double r22143 = r22141 - r22142;
        double r22144 = 3.0;
        double r22145 = pow(r22141, r22144);
        double r22146 = pow(r22142, r22144);
        double r22147 = r22145 + r22146;
        double r22148 = r22143 / r22147;
        double r22149 = r22141 * r22141;
        double r22150 = r22142 * r22142;
        double r22151 = r22141 * r22142;
        double r22152 = r22150 - r22151;
        double r22153 = r22149 + r22152;
        double r22154 = r22148 * r22153;
        double r22155 = sqrt(r22154);
        double r22156 = atan(r22155);
        double r22157 = r22140 * r22156;
        return r22157;
}

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 flip3-+0.0

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

  4. Applied associate-/r/0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))