Average Error: 0.2 → 0.2
Time: 17.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \left(1 - a\right), \left(b \cdot b\right) \cdot \left(a + 3\right)\right), 4, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\]
\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(\mathsf{fma}\left(a, a \cdot \left(1 - a\right), \left(b \cdot b\right) \cdot \left(a + 3\right)\right), 4, \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 r7921645 = a;
        double r7921646 = r7921645 * r7921645;
        double r7921647 = b;
        double r7921648 = r7921647 * r7921647;
        double r7921649 = r7921646 + r7921648;
        double r7921650 = 2.0;
        double r7921651 = pow(r7921649, r7921650);
        double r7921652 = 4.0;
        double r7921653 = 1.0;
        double r7921654 = r7921653 - r7921645;
        double r7921655 = r7921646 * r7921654;
        double r7921656 = 3.0;
        double r7921657 = r7921656 + r7921645;
        double r7921658 = r7921648 * r7921657;
        double r7921659 = r7921655 + r7921658;
        double r7921660 = r7921652 * r7921659;
        double r7921661 = r7921651 + r7921660;
        double r7921662 = r7921661 - r7921653;
        return r7921662;
}
↓
double f(double a, double b) {
        double r7921663 = a;
        double r7921664 = 1.0;
        double r7921665 = r7921664 - r7921663;
        double r7921666 = r7921663 * r7921665;
        double r7921667 = b;
        double r7921668 = r7921667 * r7921667;
        double r7921669 = 3.0;
        double r7921670 = r7921663 + r7921669;
        double r7921671 = r7921668 * r7921670;
        double r7921672 = fma(r7921663, r7921666, r7921671);
        double r7921673 = 4.0;
        double r7921674 = fma(r7921663, r7921663, r7921668);
        double r7921675 = -1.0;
        double r7921676 = fma(r7921674, r7921674, r7921675);
        double r7921677 = fma(r7921672, r7921673, r7921676);
        return r7921677;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 +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))