Average Error: 0.2 → 0.9
Time: 2.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) + {a}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) + {a}^{4}\right) - 1
double f(double a, double b) {
        double r220053 = a;
        double r220054 = r220053 * r220053;
        double r220055 = b;
        double r220056 = r220055 * r220055;
        double r220057 = r220054 + r220056;
        double r220058 = 2.0;
        double r220059 = pow(r220057, r220058);
        double r220060 = 4.0;
        double r220061 = r220060 * r220056;
        double r220062 = r220059 + r220061;
        double r220063 = 1.0;
        double r220064 = r220062 - r220063;
        return r220064;
}
↓
double f(double a, double b) {
        double r220065 = 2.0;
        double r220066 = a;
        double r220067 = pow(r220066, r220065);
        double r220068 = r220065 * r220067;
        double r220069 = b;
        double r220070 = pow(r220069, r220065);
        double r220071 = 4.0;
        double r220072 = pow(r220069, r220071);
        double r220073 = fma(r220068, r220070, r220072);
        double r220074 = pow(r220066, r220071);
        double r220075 = r220073 + r220074;
        double r220076 = 1.0;
        double r220077 = r220075 - r220076;
        return r220077;
}

Error

Bits error versus a

Bits error versus b

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. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

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