Average Error: 0.2 → 0.2
Time: 21.4s
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 r4087720 = a;
        double r4087721 = r4087720 * r4087720;
        double r4087722 = b;
        double r4087723 = r4087722 * r4087722;
        double r4087724 = r4087721 + r4087723;
        double r4087725 = 2.0;
        double r4087726 = pow(r4087724, r4087725);
        double r4087727 = 4.0;
        double r4087728 = 1.0;
        double r4087729 = r4087728 + r4087720;
        double r4087730 = r4087721 * r4087729;
        double r4087731 = 3.0;
        double r4087732 = r4087731 * r4087720;
        double r4087733 = r4087728 - r4087732;
        double r4087734 = r4087723 * r4087733;
        double r4087735 = r4087730 + r4087734;
        double r4087736 = r4087727 * r4087735;
        double r4087737 = r4087726 + r4087736;
        double r4087738 = r4087737 - r4087728;
        return r4087738;
}
↓
double f(double a, double b) {
        double r4087739 = 4.0;
        double r4087740 = a;
        double r4087741 = fma(r4087740, r4087740, r4087740);
        double r4087742 = b;
        double r4087743 = r4087742 * r4087742;
        double r4087744 = -3.0;
        double r4087745 = 1.0;
        double r4087746 = fma(r4087740, r4087744, r4087745);
        double r4087747 = r4087743 * r4087746;
        double r4087748 = fma(r4087740, r4087741, r4087747);
        double r4087749 = fma(r4087740, r4087740, r4087743);
        double r4087750 = -1.0;
        double r4087751 = fma(r4087749, r4087749, r4087750);
        double r4087752 = fma(r4087739, r4087748, r4087751);
        return r4087752;
}

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))