Average Error: 0.2 → 0.2
Time: 16.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4 \cdot b, b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(4 \cdot b, b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r248326 = a;
        double r248327 = r248326 * r248326;
        double r248328 = b;
        double r248329 = r248328 * r248328;
        double r248330 = r248327 + r248329;
        double r248331 = 2.0;
        double r248332 = pow(r248330, r248331);
        double r248333 = 4.0;
        double r248334 = r248333 * r248329;
        double r248335 = r248332 + r248334;
        double r248336 = 1.0;
        double r248337 = r248335 - r248336;
        return r248337;
}
↓
double f(double a, double b) {
        double r248338 = 4.0;
        double r248339 = b;
        double r248340 = r248338 * r248339;
        double r248341 = a;
        double r248342 = r248339 * r248339;
        double r248343 = fma(r248341, r248341, r248342);
        double r248344 = 2.0;
        double r248345 = pow(r248343, r248344);
        double r248346 = fma(r248340, r248339, r248345);
        double r248347 = 1.0;
        double r248348 = r248346 - r248347;
        return r248348;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))