Average Error: 0.2 → 0.2
Time: 25.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)\]
\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(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\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)\right)
double f(double a, double b) {
        double r10306704 = a;
        double r10306705 = r10306704 * r10306704;
        double r10306706 = b;
        double r10306707 = r10306706 * r10306706;
        double r10306708 = r10306705 + r10306707;
        double r10306709 = 2.0;
        double r10306710 = pow(r10306708, r10306709);
        double r10306711 = 4.0;
        double r10306712 = 1.0;
        double r10306713 = r10306712 + r10306704;
        double r10306714 = r10306705 * r10306713;
        double r10306715 = 3.0;
        double r10306716 = r10306715 * r10306704;
        double r10306717 = r10306712 - r10306716;
        double r10306718 = r10306707 * r10306717;
        double r10306719 = r10306714 + r10306718;
        double r10306720 = r10306711 * r10306719;
        double r10306721 = r10306710 + r10306720;
        double r10306722 = r10306721 - r10306712;
        return r10306722;
}
↓
double f(double a, double b) {
        double r10306723 = 4.0;
        double r10306724 = a;
        double r10306725 = fma(r10306723, r10306724, r10306723);
        double r10306726 = r10306724 * r10306724;
        double r10306727 = b;
        double r10306728 = r10306727 * r10306727;
        double r10306729 = -12.0;
        double r10306730 = fma(r10306724, r10306729, r10306723);
        double r10306731 = fma(r10306724, r10306724, r10306728);
        double r10306732 = -1.0;
        double r10306733 = fma(r10306731, r10306731, r10306732);
        double r10306734 = fma(r10306728, r10306730, r10306733);
        double r10306735 = fma(r10306725, r10306726, r10306734);
        return r10306735;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)\]

Reproduce

herbie shell --seed 2019158 +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))