Average Error: 0.2 → 1.0
Time: 28.9s
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 r180544 = a;
        double r180545 = r180544 * r180544;
        double r180546 = b;
        double r180547 = r180546 * r180546;
        double r180548 = r180545 + r180547;
        double r180549 = 2.0;
        double r180550 = pow(r180548, r180549);
        double r180551 = 4.0;
        double r180552 = r180551 * r180547;
        double r180553 = r180550 + r180552;
        double r180554 = 1.0;
        double r180555 = r180553 - r180554;
        return r180555;
}
↓
double f(double a, double b) {
        double r180556 = a;
        double r180557 = 4.0;
        double r180558 = pow(r180556, r180557);
        double r180559 = b;
        double r180560 = pow(r180559, r180557);
        double r180561 = 2.0;
        double r180562 = pow(r180556, r180561);
        double r180563 = pow(r180559, r180561);
        double r180564 = r180562 * r180563;
        double r180565 = r180561 * r180564;
        double r180566 = r180560 + r180565;
        double r180567 = r180558 + r180566;
        double r180568 = 1.0;
        double r180569 = r180567 - r180568;
        return r180569;
}

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))