Average Error: 0.2 → 1.0
Time: 10.8s
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 r227651 = a;
        double r227652 = r227651 * r227651;
        double r227653 = b;
        double r227654 = r227653 * r227653;
        double r227655 = r227652 + r227654;
        double r227656 = 2.0;
        double r227657 = pow(r227655, r227656);
        double r227658 = 4.0;
        double r227659 = r227658 * r227654;
        double r227660 = r227657 + r227659;
        double r227661 = 1.0;
        double r227662 = r227660 - r227661;
        return r227662;
}
↓
double f(double a, double b) {
        double r227663 = a;
        double r227664 = 4.0;
        double r227665 = pow(r227663, r227664);
        double r227666 = b;
        double r227667 = pow(r227666, r227664);
        double r227668 = 2.0;
        double r227669 = pow(r227663, r227668);
        double r227670 = pow(r227666, r227668);
        double r227671 = r227669 * r227670;
        double r227672 = r227668 * r227671;
        double r227673 = r227667 + r227672;
        double r227674 = r227665 + r227673;
        double r227675 = 1.0;
        double r227676 = r227674 - r227675;
        return r227676;
}

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 2020047 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))