Average Error: 0.2 → 0.2
Time: 26.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\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)\right)
double f(double a, double b) {
        double r17242499 = a;
        double r17242500 = r17242499 * r17242499;
        double r17242501 = b;
        double r17242502 = r17242501 * r17242501;
        double r17242503 = r17242500 + r17242502;
        double r17242504 = 2.0;
        double r17242505 = pow(r17242503, r17242504);
        double r17242506 = 4.0;
        double r17242507 = 1.0;
        double r17242508 = r17242507 + r17242499;
        double r17242509 = r17242500 * r17242508;
        double r17242510 = 3.0;
        double r17242511 = r17242510 * r17242499;
        double r17242512 = r17242507 - r17242511;
        double r17242513 = r17242502 * r17242512;
        double r17242514 = r17242509 + r17242513;
        double r17242515 = r17242506 * r17242514;
        double r17242516 = r17242505 + r17242515;
        double r17242517 = r17242516 - r17242507;
        return r17242517;
}
↓
double f(double a, double b) {
        double r17242518 = 4.0;
        double r17242519 = a;
        double r17242520 = fma(r17242518, r17242519, r17242518);
        double r17242521 = r17242519 * r17242519;
        double r17242522 = b;
        double r17242523 = r17242522 * r17242522;
        double r17242524 = -12.0;
        double r17242525 = fma(r17242519, r17242524, r17242518);
        double r17242526 = fma(r17242519, r17242519, r17242523);
        double r17242527 = -1.0;
        double r17242528 = fma(r17242526, r17242526, r17242527);
        double r17242529 = fma(r17242523, r17242525, r17242528);
        double r17242530 = fma(r17242520, r17242521, r17242529);
        return r17242530;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(4, a, 4\right), a \cdot a, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(a, -12, 4\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\right)\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))