Average Error: 0.2 → 0.9
Time: 18.8s
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 r156956 = a;
        double r156957 = r156956 * r156956;
        double r156958 = b;
        double r156959 = r156958 * r156958;
        double r156960 = r156957 + r156959;
        double r156961 = 2.0;
        double r156962 = pow(r156960, r156961);
        double r156963 = 4.0;
        double r156964 = r156963 * r156959;
        double r156965 = r156962 + r156964;
        double r156966 = 1.0;
        double r156967 = r156965 - r156966;
        return r156967;
}
↓
double f(double a, double b) {
        double r156968 = a;
        double r156969 = 4.0;
        double r156970 = pow(r156968, r156969);
        double r156971 = b;
        double r156972 = pow(r156971, r156969);
        double r156973 = 2.0;
        double r156974 = pow(r156968, r156973);
        double r156975 = pow(r156971, r156973);
        double r156976 = r156974 * r156975;
        double r156977 = r156973 * r156976;
        double r156978 = r156972 + r156977;
        double r156979 = r156970 + r156978;
        double r156980 = 1.0;
        double r156981 = r156979 - r156980;
        return r156981;
}

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 2019235 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))