Average Error: 0.2 → 1.0
Time: 3.2s
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 r308126 = a;
        double r308127 = r308126 * r308126;
        double r308128 = b;
        double r308129 = r308128 * r308128;
        double r308130 = r308127 + r308129;
        double r308131 = 2.0;
        double r308132 = pow(r308130, r308131);
        double r308133 = 4.0;
        double r308134 = r308133 * r308129;
        double r308135 = r308132 + r308134;
        double r308136 = 1.0;
        double r308137 = r308135 - r308136;
        return r308137;
}
↓
double f(double a, double b) {
        double r308138 = a;
        double r308139 = 4.0;
        double r308140 = pow(r308138, r308139);
        double r308141 = b;
        double r308142 = pow(r308141, r308139);
        double r308143 = 2.0;
        double r308144 = pow(r308138, r308143);
        double r308145 = pow(r308141, r308143);
        double r308146 = r308144 * r308145;
        double r308147 = r308143 * r308146;
        double r308148 = r308142 + r308147;
        double r308149 = r308140 + r308148;
        double r308150 = 1.0;
        double r308151 = r308149 - r308150;
        return r308151;
}

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 2020047 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))