Average Error: 0.2 → 0.9
Time: 2.7s
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 r288116 = a;
        double r288117 = r288116 * r288116;
        double r288118 = b;
        double r288119 = r288118 * r288118;
        double r288120 = r288117 + r288119;
        double r288121 = 2.0;
        double r288122 = pow(r288120, r288121);
        double r288123 = 4.0;
        double r288124 = r288123 * r288119;
        double r288125 = r288122 + r288124;
        double r288126 = 1.0;
        double r288127 = r288125 - r288126;
        return r288127;
}
↓
double f(double a, double b) {
        double r288128 = a;
        double r288129 = 4.0;
        double r288130 = pow(r288128, r288129);
        double r288131 = b;
        double r288132 = pow(r288131, r288129);
        double r288133 = 2.0;
        double r288134 = pow(r288128, r288133);
        double r288135 = pow(r288131, r288133);
        double r288136 = r288134 * r288135;
        double r288137 = r288133 * r288136;
        double r288138 = r288132 + r288137;
        double r288139 = r288130 + r288138;
        double r288140 = 1.0;
        double r288141 = r288139 - r288140;
        return r288141;
}

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 2020062 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))