Average Error: 0.2 → 0.9
Time: 3.7s
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 r174241 = a;
        double r174242 = r174241 * r174241;
        double r174243 = b;
        double r174244 = r174243 * r174243;
        double r174245 = r174242 + r174244;
        double r174246 = 2.0;
        double r174247 = pow(r174245, r174246);
        double r174248 = 4.0;
        double r174249 = r174248 * r174244;
        double r174250 = r174247 + r174249;
        double r174251 = 1.0;
        double r174252 = r174250 - r174251;
        return r174252;
}
↓
double f(double a, double b) {
        double r174253 = a;
        double r174254 = 4.0;
        double r174255 = pow(r174253, r174254);
        double r174256 = b;
        double r174257 = pow(r174256, r174254);
        double r174258 = 2.0;
        double r174259 = pow(r174253, r174258);
        double r174260 = pow(r174256, r174258);
        double r174261 = r174259 * r174260;
        double r174262 = r174258 * r174261;
        double r174263 = r174257 + r174262;
        double r174264 = r174255 + r174263;
        double r174265 = 1.0;
        double r174266 = r174264 - r174265;
        return r174266;
}

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 2020083 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))