Average Error: 0.2 → 0.2
Time: 22.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)
double f(double a, double b) {
        double r6393942 = a;
        double r6393943 = r6393942 * r6393942;
        double r6393944 = b;
        double r6393945 = r6393944 * r6393944;
        double r6393946 = r6393943 + r6393945;
        double r6393947 = 2.0;
        double r6393948 = pow(r6393946, r6393947);
        double r6393949 = 4.0;
        double r6393950 = 1.0;
        double r6393951 = r6393950 - r6393942;
        double r6393952 = r6393943 * r6393951;
        double r6393953 = 3.0;
        double r6393954 = r6393953 + r6393942;
        double r6393955 = r6393945 * r6393954;
        double r6393956 = r6393952 + r6393955;
        double r6393957 = r6393949 * r6393956;
        double r6393958 = r6393948 + r6393957;
        double r6393959 = r6393958 - r6393950;
        return r6393959;
}
↓
double f(double a, double b) {
        double r6393960 = 4.0;
        double r6393961 = a;
        double r6393962 = r6393961 * r6393961;
        double r6393963 = r6393961 - r6393962;
        double r6393964 = b;
        double r6393965 = r6393964 * r6393964;
        double r6393966 = 3.0;
        double r6393967 = r6393961 + r6393966;
        double r6393968 = r6393965 * r6393967;
        double r6393969 = fma(r6393961, r6393963, r6393968);
        double r6393970 = fma(r6393961, r6393961, r6393965);
        double r6393971 = -1.0;
        double r6393972 = fma(r6393970, r6393970, r6393971);
        double r6393973 = fma(r6393960, r6393969, r6393972);
        return r6393973;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(a + 3\right) \cdot \left(b \cdot b\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))