Average Error: 0.2 → 1.0
Time: 18.7s
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 r237657 = a;
        double r237658 = r237657 * r237657;
        double r237659 = b;
        double r237660 = r237659 * r237659;
        double r237661 = r237658 + r237660;
        double r237662 = 2.0;
        double r237663 = pow(r237661, r237662);
        double r237664 = 4.0;
        double r237665 = r237664 * r237660;
        double r237666 = r237663 + r237665;
        double r237667 = 1.0;
        double r237668 = r237666 - r237667;
        return r237668;
}
↓
double f(double a, double b) {
        double r237669 = a;
        double r237670 = 4.0;
        double r237671 = pow(r237669, r237670);
        double r237672 = b;
        double r237673 = pow(r237672, r237670);
        double r237674 = 2.0;
        double r237675 = pow(r237669, r237674);
        double r237676 = pow(r237672, r237674);
        double r237677 = r237675 * r237676;
        double r237678 = r237674 * r237677;
        double r237679 = r237673 + r237678;
        double r237680 = r237671 + r237679;
        double r237681 = 1.0;
        double r237682 = r237680 - r237681;
        return r237682;
}

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))