Average Error: 0.2 → 0.2
Time: 22.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(3, b \cdot b, a \cdot \mathsf{fma}\left(1 - a, a, b \cdot b\right)\right), 4, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(\mathsf{fma}\left(3, b \cdot b, a \cdot \mathsf{fma}\left(1 - a, a, b \cdot b\right)\right), 4, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r10532693 = a;
        double r10532694 = r10532693 * r10532693;
        double r10532695 = b;
        double r10532696 = r10532695 * r10532695;
        double r10532697 = r10532694 + r10532696;
        double r10532698 = 2.0;
        double r10532699 = pow(r10532697, r10532698);
        double r10532700 = 4.0;
        double r10532701 = 1.0;
        double r10532702 = r10532701 - r10532693;
        double r10532703 = r10532694 * r10532702;
        double r10532704 = 3.0;
        double r10532705 = r10532704 + r10532693;
        double r10532706 = r10532696 * r10532705;
        double r10532707 = r10532703 + r10532706;
        double r10532708 = r10532700 * r10532707;
        double r10532709 = r10532699 + r10532708;
        double r10532710 = r10532709 - r10532701;
        return r10532710;
}
↓
double f(double a, double b) {
        double r10532711 = 3.0;
        double r10532712 = b;
        double r10532713 = r10532712 * r10532712;
        double r10532714 = a;
        double r10532715 = 1.0;
        double r10532716 = r10532715 - r10532714;
        double r10532717 = fma(r10532716, r10532714, r10532713);
        double r10532718 = r10532714 * r10532717;
        double r10532719 = fma(r10532711, r10532713, r10532718);
        double r10532720 = 4.0;
        double r10532721 = r10532714 * r10532714;
        double r10532722 = fma(r10532712, r10532712, r10532721);
        double r10532723 = 2.0;
        double r10532724 = pow(r10532722, r10532723);
        double r10532725 = fma(r10532719, r10532720, r10532724);
        double r10532726 = r10532725 - r10532715;
        return r10532726;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))