Average Error: 0.2 → 0.2
Time: 19.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\]
\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(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\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)
double f(double a, double b) {
        double r5640623 = a;
        double r5640624 = r5640623 * r5640623;
        double r5640625 = b;
        double r5640626 = r5640625 * r5640625;
        double r5640627 = r5640624 + r5640626;
        double r5640628 = 2.0;
        double r5640629 = pow(r5640627, r5640628);
        double r5640630 = 4.0;
        double r5640631 = 1.0;
        double r5640632 = r5640631 - r5640623;
        double r5640633 = r5640624 * r5640632;
        double r5640634 = 3.0;
        double r5640635 = r5640634 + r5640623;
        double r5640636 = r5640626 * r5640635;
        double r5640637 = r5640633 + r5640636;
        double r5640638 = r5640630 * r5640637;
        double r5640639 = r5640629 + r5640638;
        double r5640640 = r5640639 - r5640631;
        return r5640640;
}
↓
double f(double a, double b) {
        double r5640641 = 4.0;
        double r5640642 = a;
        double r5640643 = r5640642 * r5640642;
        double r5640644 = r5640642 - r5640643;
        double r5640645 = b;
        double r5640646 = r5640645 * r5640645;
        double r5640647 = 3.0;
        double r5640648 = r5640642 + r5640647;
        double r5640649 = r5640646 * r5640648;
        double r5640650 = fma(r5640642, r5640644, r5640649);
        double r5640651 = fma(r5640642, r5640642, r5640646);
        double r5640652 = -1.0;
        double r5640653 = fma(r5640651, r5640651, r5640652);
        double r5640654 = fma(r5640641, r5640650, r5640653);
        return r5640654;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019146 +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))