Average Error: 0.2 → 0.2
Time: 22.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, -3, 1\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a, \mathsf{fma}\left(a, a, a\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(a, -3, 1\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 r3434775 = a;
        double r3434776 = r3434775 * r3434775;
        double r3434777 = b;
        double r3434778 = r3434777 * r3434777;
        double r3434779 = r3434776 + r3434778;
        double r3434780 = 2.0;
        double r3434781 = pow(r3434779, r3434780);
        double r3434782 = 4.0;
        double r3434783 = 1.0;
        double r3434784 = r3434783 + r3434775;
        double r3434785 = r3434776 * r3434784;
        double r3434786 = 3.0;
        double r3434787 = r3434786 * r3434775;
        double r3434788 = r3434783 - r3434787;
        double r3434789 = r3434778 * r3434788;
        double r3434790 = r3434785 + r3434789;
        double r3434791 = r3434782 * r3434790;
        double r3434792 = r3434781 + r3434791;
        double r3434793 = r3434792 - r3434783;
        return r3434793;
}
↓
double f(double a, double b) {
        double r3434794 = 4.0;
        double r3434795 = a;
        double r3434796 = fma(r3434795, r3434795, r3434795);
        double r3434797 = b;
        double r3434798 = r3434797 * r3434797;
        double r3434799 = -3.0;
        double r3434800 = 1.0;
        double r3434801 = fma(r3434795, r3434799, r3434800);
        double r3434802 = r3434798 * r3434801;
        double r3434803 = fma(r3434795, r3434796, r3434802);
        double r3434804 = fma(r3434795, r3434795, r3434798);
        double r3434805 = -1.0;
        double r3434806 = fma(r3434804, r3434804, r3434805);
        double r3434807 = fma(r3434794, r3434803, r3434806);
        return r3434807;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))