Average Error: 0.2 → 1.0
Time: 10.2s
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 r250719 = a;
        double r250720 = r250719 * r250719;
        double r250721 = b;
        double r250722 = r250721 * r250721;
        double r250723 = r250720 + r250722;
        double r250724 = 2.0;
        double r250725 = pow(r250723, r250724);
        double r250726 = 4.0;
        double r250727 = r250726 * r250722;
        double r250728 = r250725 + r250727;
        double r250729 = 1.0;
        double r250730 = r250728 - r250729;
        return r250730;
}
↓
double f(double a, double b) {
        double r250731 = a;
        double r250732 = 4.0;
        double r250733 = pow(r250731, r250732);
        double r250734 = b;
        double r250735 = pow(r250734, r250732);
        double r250736 = 2.0;
        double r250737 = pow(r250731, r250736);
        double r250738 = pow(r250734, r250736);
        double r250739 = r250737 * r250738;
        double r250740 = r250736 * r250739;
        double r250741 = r250735 + r250740;
        double r250742 = r250733 + r250741;
        double r250743 = 1.0;
        double r250744 = r250742 - r250743;
        return r250744;
}

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 2019294 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))