Average Error: 0.2 → 1.0
Time: 9.9s
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 r155580 = a;
        double r155581 = r155580 * r155580;
        double r155582 = b;
        double r155583 = r155582 * r155582;
        double r155584 = r155581 + r155583;
        double r155585 = 2.0;
        double r155586 = pow(r155584, r155585);
        double r155587 = 4.0;
        double r155588 = r155587 * r155583;
        double r155589 = r155586 + r155588;
        double r155590 = 1.0;
        double r155591 = r155589 - r155590;
        return r155591;
}
↓
double f(double a, double b) {
        double r155592 = a;
        double r155593 = 4.0;
        double r155594 = pow(r155592, r155593);
        double r155595 = b;
        double r155596 = pow(r155595, r155593);
        double r155597 = 2.0;
        double r155598 = pow(r155592, r155597);
        double r155599 = pow(r155595, r155597);
        double r155600 = r155598 * r155599;
        double r155601 = r155597 * r155600;
        double r155602 = r155596 + r155601;
        double r155603 = r155594 + r155602;
        double r155604 = 1.0;
        double r155605 = r155603 - r155604;
        return r155605;
}

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))