Average Error: 0.2 → 0.2
Time: 34.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(b, 4 \cdot b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(b, 4 \cdot b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r302951 = a;
        double r302952 = r302951 * r302951;
        double r302953 = b;
        double r302954 = r302953 * r302953;
        double r302955 = r302952 + r302954;
        double r302956 = 2.0;
        double r302957 = pow(r302955, r302956);
        double r302958 = 4.0;
        double r302959 = r302958 * r302954;
        double r302960 = r302957 + r302959;
        double r302961 = 1.0;
        double r302962 = r302960 - r302961;
        return r302962;
}
↓
double f(double a, double b) {
        double r302963 = b;
        double r302964 = 4.0;
        double r302965 = r302964 * r302963;
        double r302966 = a;
        double r302967 = r302963 * r302963;
        double r302968 = fma(r302966, r302966, r302967);
        double r302969 = 2.0;
        double r302970 = pow(r302968, r302969);
        double r302971 = fma(r302963, r302965, r302970);
        double r302972 = 1.0;
        double r302973 = r302971 - r302972;
        return r302973;
}

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. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))