Average Error: 0.2 → 0.9
Time: 15.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1
double f(double a, double b) {
        double r100882 = a;
        double r100883 = r100882 * r100882;
        double r100884 = b;
        double r100885 = r100884 * r100884;
        double r100886 = r100883 + r100885;
        double r100887 = 2.0;
        double r100888 = pow(r100886, r100887);
        double r100889 = 4.0;
        double r100890 = r100889 * r100885;
        double r100891 = r100888 + r100890;
        double r100892 = 1.0;
        double r100893 = r100891 - r100892;
        return r100893;
}
↓
double f(double a, double b) {
        double r100894 = a;
        double r100895 = 4.0;
        double r100896 = pow(r100894, r100895);
        double r100897 = b;
        double r100898 = pow(r100897, r100895);
        double r100899 = 2.0;
        double r100900 = pow(r100894, r100899);
        double r100901 = pow(r100897, r100899);
        double r100902 = r100900 * r100901;
        double r100903 = r100899 * r100902;
        double r100904 = r100898 + r100903;
        double r100905 = r100896 + r100904;
        double r100906 = 1.0;
        double r100907 = r100905 - r100906;
        return r100907;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  2. Taylor expanded around inf 0.9

    \[\leadsto \color{blue}{\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} - 1\]

  3. Final simplification0.9

    \[\leadsto \left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))