Average Error: 0.2 → 0.8
Time: 2.9s
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 r310912 = a;
        double r310913 = r310912 * r310912;
        double r310914 = b;
        double r310915 = r310914 * r310914;
        double r310916 = r310913 + r310915;
        double r310917 = 2.0;
        double r310918 = pow(r310916, r310917);
        double r310919 = 4.0;
        double r310920 = r310919 * r310915;
        double r310921 = r310918 + r310920;
        double r310922 = 1.0;
        double r310923 = r310921 - r310922;
        return r310923;
}
↓
double f(double a, double b) {
        double r310924 = a;
        double r310925 = 4.0;
        double r310926 = pow(r310924, r310925);
        double r310927 = b;
        double r310928 = pow(r310927, r310925);
        double r310929 = 2.0;
        double r310930 = pow(r310924, r310929);
        double r310931 = pow(r310927, r310929);
        double r310932 = r310930 * r310931;
        double r310933 = r310929 * r310932;
        double r310934 = r310928 + r310933;
        double r310935 = r310926 + r310934;
        double r310936 = 1.0;
        double r310937 = r310935 - r310936;
        return r310937;
}

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 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020059 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))