Average Error: 0.2 → 0.9
Time: 10.4s
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 r167843 = a;
        double r167844 = r167843 * r167843;
        double r167845 = b;
        double r167846 = r167845 * r167845;
        double r167847 = r167844 + r167846;
        double r167848 = 2.0;
        double r167849 = pow(r167847, r167848);
        double r167850 = 4.0;
        double r167851 = r167850 * r167846;
        double r167852 = r167849 + r167851;
        double r167853 = 1.0;
        double r167854 = r167852 - r167853;
        return r167854;
}
↓
double f(double a, double b) {
        double r167855 = a;
        double r167856 = 4.0;
        double r167857 = pow(r167855, r167856);
        double r167858 = b;
        double r167859 = pow(r167858, r167856);
        double r167860 = 2.0;
        double r167861 = pow(r167855, r167860);
        double r167862 = pow(r167858, r167860);
        double r167863 = r167861 * r167862;
        double r167864 = r167860 * r167863;
        double r167865 = r167859 + r167864;
        double r167866 = r167857 + r167865;
        double r167867 = 1.0;
        double r167868 = r167866 - r167867;
        return r167868;
}

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))