Average Error: 0.2 → 0.2
Time: 23.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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 \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r103684 = a;
        double r103685 = r103684 * r103684;
        double r103686 = b;
        double r103687 = r103686 * r103686;
        double r103688 = r103685 + r103687;
        double r103689 = 2.0;
        double r103690 = pow(r103688, r103689);
        double r103691 = 4.0;
        double r103692 = 1.0;
        double r103693 = r103692 + r103684;
        double r103694 = r103685 * r103693;
        double r103695 = 3.0;
        double r103696 = r103695 * r103684;
        double r103697 = r103692 - r103696;
        double r103698 = r103687 * r103697;
        double r103699 = r103694 + r103698;
        double r103700 = r103691 * r103699;
        double r103701 = r103690 + r103700;
        double r103702 = r103701 - r103692;
        return r103702;
}
↓
double f(double a, double b) {
        double r103703 = 4.0;
        double r103704 = a;
        double r103705 = r103704 * r103704;
        double r103706 = 1.0;
        double r103707 = r103706 + r103704;
        double r103708 = b;
        double r103709 = r103708 * r103708;
        double r103710 = 3.0;
        double r103711 = r103710 * r103704;
        double r103712 = r103706 - r103711;
        double r103713 = r103709 * r103712;
        double r103714 = fma(r103705, r103707, r103713);
        double r103715 = fma(r103704, r103704, r103709);
        double r103716 = 2.0;
        double r103717 = pow(r103715, r103716);
        double r103718 = fma(r103703, r103714, r103717);
        double r103719 = r103718 - r103706;
        return r103719;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1}\]

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))