Average Error: 0.2 → 0.9
Time: 4.6s
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 r284679 = a;
        double r284680 = r284679 * r284679;
        double r284681 = b;
        double r284682 = r284681 * r284681;
        double r284683 = r284680 + r284682;
        double r284684 = 2.0;
        double r284685 = pow(r284683, r284684);
        double r284686 = 4.0;
        double r284687 = r284686 * r284682;
        double r284688 = r284685 + r284687;
        double r284689 = 1.0;
        double r284690 = r284688 - r284689;
        return r284690;
}
↓
double f(double a, double b) {
        double r284691 = a;
        double r284692 = 4.0;
        double r284693 = pow(r284691, r284692);
        double r284694 = b;
        double r284695 = pow(r284694, r284692);
        double r284696 = 2.0;
        double r284697 = pow(r284691, r284696);
        double r284698 = pow(r284694, r284696);
        double r284699 = r284697 * r284698;
        double r284700 = r284696 * r284699;
        double r284701 = r284695 + r284700;
        double r284702 = r284693 + r284701;
        double r284703 = 1.0;
        double r284704 = r284702 - r284703;
        return r284704;
}

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 2020089 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))