Average Error: 0.2 → 0.9
Time: 15.6s
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 r99909 = a;
        double r99910 = r99909 * r99909;
        double r99911 = b;
        double r99912 = r99911 * r99911;
        double r99913 = r99910 + r99912;
        double r99914 = 2.0;
        double r99915 = pow(r99913, r99914);
        double r99916 = 4.0;
        double r99917 = r99916 * r99912;
        double r99918 = r99915 + r99917;
        double r99919 = 1.0;
        double r99920 = r99918 - r99919;
        return r99920;
}
↓
double f(double a, double b) {
        double r99921 = a;
        double r99922 = 4.0;
        double r99923 = pow(r99921, r99922);
        double r99924 = b;
        double r99925 = pow(r99924, r99922);
        double r99926 = 2.0;
        double r99927 = pow(r99921, r99926);
        double r99928 = pow(r99924, r99926);
        double r99929 = r99927 * r99928;
        double r99930 = r99926 * r99929;
        double r99931 = r99925 + r99930;
        double r99932 = r99923 + r99931;
        double r99933 = 1.0;
        double r99934 = r99932 - r99933;
        return r99934;
}

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 0.9

    \[\leadsto \color{blue}{\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} - 1\]

  3. Final simplification0.9

    \[\leadsto \left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))