Average Error: 0.2 → 0.2
Time: 54.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\left(b \cdot b\right), 3, \left(a \cdot a\right)\right) - a \cdot \left(a \cdot a\right)\right), \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b, b, \left(a \cdot a\right)\right)\right), \left(\mathsf{fma}\left(b, b, \left(a \cdot a\right)\right)\right), -1\right)\right)\right)\]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \left(\mathsf{fma}\left(\left(b \cdot b\right), 3, \left(a \cdot a\right)\right) - a \cdot \left(a \cdot a\right)\right), \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(b, b, \left(a \cdot a\right)\right)\right), \left(\mathsf{fma}\left(b, b, \left(a \cdot a\right)\right)\right), -1\right)\right)\right)
double f(double a, double b) {
        double r63043506 = a;
        double r63043507 = r63043506 * r63043506;
        double r63043508 = b;
        double r63043509 = r63043508 * r63043508;
        double r63043510 = r63043507 + r63043509;
        double r63043511 = 2.0;
        double r63043512 = pow(r63043510, r63043511);
        double r63043513 = 4.0;
        double r63043514 = 1.0;
        double r63043515 = r63043514 - r63043506;
        double r63043516 = r63043507 * r63043515;
        double r63043517 = 3.0;
        double r63043518 = r63043517 + r63043506;
        double r63043519 = r63043509 * r63043518;
        double r63043520 = r63043516 + r63043519;
        double r63043521 = r63043513 * r63043520;
        double r63043522 = r63043512 + r63043521;
        double r63043523 = r63043522 - r63043514;
        return r63043523;
}
↓
double f(double a, double b) {
        double r63043524 = 4.0;
        double r63043525 = b;
        double r63043526 = r63043525 * r63043525;
        double r63043527 = 3.0;
        double r63043528 = a;
        double r63043529 = r63043528 * r63043528;
        double r63043530 = fma(r63043526, r63043527, r63043529);
        double r63043531 = r63043528 * r63043529;
        double r63043532 = r63043530 - r63043531;
        double r63043533 = fma(r63043525, r63043525, r63043529);
        double r63043534 = -1.0;
        double r63043535 = fma(r63043533, r63043533, r63043534);
        double r63043536 = fma(r63043524, r63043532, r63043535);
        return r63043536;
}

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

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))