Average Error: 0.0 → 0.0
Time: 3.6s
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 r11125 = 2.0;
        double r11126 = 1.0;
        double r11127 = x;
        double r11128 = r11126 - r11127;
        double r11129 = r11126 + r11127;
        double r11130 = r11128 / r11129;
        double r11131 = sqrt(r11130);
        double r11132 = atan(r11131);
        double r11133 = r11125 * r11132;
        return r11133;
}
↓
double f(double x) {
        double r11134 = 2.0;
        double r11135 = 1.0;
        double r11136 = x;
        double r11137 = r11135 - r11136;
        double r11138 = 3.0;
        double r11139 = pow(r11135, r11138);
        double r11140 = pow(r11136, r11138);
        double r11141 = r11139 + r11140;
        double r11142 = r11137 / r11141;
        double r11143 = r11135 * r11135;
        double r11144 = r11136 * r11136;
        double r11145 = r11135 * r11136;
        double r11146 = r11144 - r11145;
        double r11147 = r11143 + r11146;
        double r11148 = r11142 * r11147;
        double r11149 = sqrt(r11148);
        double r11150 = atan(r11149);
        double r11151 = r11134 * r11150;
        return r11151;
}

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 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 2020001 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))