Average Error: 0.2 → 0.2
Time: 38.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(b \cdot 4, b, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(b \cdot 4, b, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r8448705 = a;
        double r8448706 = r8448705 * r8448705;
        double r8448707 = b;
        double r8448708 = r8448707 * r8448707;
        double r8448709 = r8448706 + r8448708;
        double r8448710 = 2.0;
        double r8448711 = pow(r8448709, r8448710);
        double r8448712 = 4.0;
        double r8448713 = r8448712 * r8448708;
        double r8448714 = r8448711 + r8448713;
        double r8448715 = 1.0;
        double r8448716 = r8448714 - r8448715;
        return r8448716;
}
↓
double f(double a, double b) {
        double r8448717 = b;
        double r8448718 = 4.0;
        double r8448719 = r8448717 * r8448718;
        double r8448720 = a;
        double r8448721 = r8448720 * r8448720;
        double r8448722 = fma(r8448717, r8448717, r8448721);
        double r8448723 = 2.0;
        double r8448724 = pow(r8448722, r8448723);
        double r8448725 = fma(r8448719, r8448717, r8448724);
        double r8448726 = 1.0;
        double r8448727 = r8448725 - r8448726;
        return r8448727;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))