Average Error: 0.2 → 1.0
Time: 18.5s
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 r240059 = a;
        double r240060 = r240059 * r240059;
        double r240061 = b;
        double r240062 = r240061 * r240061;
        double r240063 = r240060 + r240062;
        double r240064 = 2.0;
        double r240065 = pow(r240063, r240064);
        double r240066 = 4.0;
        double r240067 = r240066 * r240062;
        double r240068 = r240065 + r240067;
        double r240069 = 1.0;
        double r240070 = r240068 - r240069;
        return r240070;
}
↓
double f(double a, double b) {
        double r240071 = a;
        double r240072 = 4.0;
        double r240073 = pow(r240071, r240072);
        double r240074 = b;
        double r240075 = pow(r240074, r240072);
        double r240076 = 2.0;
        double r240077 = pow(r240071, r240076);
        double r240078 = pow(r240074, r240076);
        double r240079 = r240077 * r240078;
        double r240080 = r240076 * r240079;
        double r240081 = r240075 + r240080;
        double r240082 = r240073 + r240081;
        double r240083 = 1.0;
        double r240084 = r240082 - r240083;
        return r240084;
}

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 1.0

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

  3. Final simplification1.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))