Average Error: 0.2 → 0.2
Time: 5.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot 3, b, \left(a \cdot a\right) \cdot \left(1 - a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)\]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot 3, b, \left(a \cdot a\right) \cdot \left(1 - a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)
double f(double a, double b) {
        double r394602 = a;
        double r394603 = r394602 * r394602;
        double r394604 = b;
        double r394605 = r394604 * r394604;
        double r394606 = r394603 + r394605;
        double r394607 = 2.0;
        double r394608 = pow(r394606, r394607);
        double r394609 = 4.0;
        double r394610 = 1.0;
        double r394611 = r394610 - r394602;
        double r394612 = r394603 * r394611;
        double r394613 = 3.0;
        double r394614 = r394613 + r394602;
        double r394615 = r394605 * r394614;
        double r394616 = r394612 + r394615;
        double r394617 = r394609 * r394616;
        double r394618 = r394608 + r394617;
        double r394619 = r394618 - r394610;
        return r394619;
}
↓
double f(double a, double b) {
        double r394620 = 4.0;
        double r394621 = b;
        double r394622 = 3.0;
        double r394623 = r394621 * r394622;
        double r394624 = a;
        double r394625 = r394624 * r394624;
        double r394626 = 1.0;
        double r394627 = r394626 - r394624;
        double r394628 = r394625 * r394627;
        double r394629 = fma(r394623, r394621, r394628);
        double r394630 = r394621 * r394621;
        double r394631 = r394625 + r394630;
        double r394632 = 2.0;
        double r394633 = pow(r394631, r394632);
        double r394634 = r394633 - r394626;
        double r394635 = fma(r394620, r394629, r394634);
        return r394635;
}

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

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))