Average Error: 0.2 → 0.2
Time: 28.5s
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 r215250 = a;
        double r215251 = r215250 * r215250;
        double r215252 = b;
        double r215253 = r215252 * r215252;
        double r215254 = r215251 + r215253;
        double r215255 = 2.0;
        double r215256 = pow(r215254, r215255);
        double r215257 = 4.0;
        double r215258 = r215257 * r215253;
        double r215259 = r215256 + r215258;
        double r215260 = 1.0;
        double r215261 = r215259 - r215260;
        return r215261;
}
↓
double f(double a, double b) {
        double r215262 = 4.0;
        double r215263 = b;
        double r215264 = r215262 * r215263;
        double r215265 = a;
        double r215266 = r215263 * r215263;
        double r215267 = fma(r215265, r215265, r215266);
        double r215268 = 2.0;
        double r215269 = pow(r215267, r215268);
        double r215270 = fma(r215264, r215263, r215269);
        double r215271 = 1.0;
        double r215272 = r215270 - r215271;
        return r215272;
}

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 2019199 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))