Average Error: 0.2 → 0.2
Time: 21.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \left(1 - a\right), \left(b \cdot b\right) \cdot \left(a + 3\right)\right), 4, \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\]
\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(\mathsf{fma}\left(a, a \cdot \left(1 - a\right), \left(b \cdot b\right) \cdot \left(a + 3\right)\right), 4, \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 r3943237 = a;
        double r3943238 = r3943237 * r3943237;
        double r3943239 = b;
        double r3943240 = r3943239 * r3943239;
        double r3943241 = r3943238 + r3943240;
        double r3943242 = 2.0;
        double r3943243 = pow(r3943241, r3943242);
        double r3943244 = 4.0;
        double r3943245 = 1.0;
        double r3943246 = r3943245 - r3943237;
        double r3943247 = r3943238 * r3943246;
        double r3943248 = 3.0;
        double r3943249 = r3943248 + r3943237;
        double r3943250 = r3943240 * r3943249;
        double r3943251 = r3943247 + r3943250;
        double r3943252 = r3943244 * r3943251;
        double r3943253 = r3943243 + r3943252;
        double r3943254 = r3943253 - r3943245;
        return r3943254;
}
↓
double f(double a, double b) {
        double r3943255 = a;
        double r3943256 = 1.0;
        double r3943257 = r3943256 - r3943255;
        double r3943258 = r3943255 * r3943257;
        double r3943259 = b;
        double r3943260 = r3943259 * r3943259;
        double r3943261 = 3.0;
        double r3943262 = r3943255 + r3943261;
        double r3943263 = r3943260 * r3943262;
        double r3943264 = fma(r3943255, r3943258, r3943263);
        double r3943265 = 4.0;
        double r3943266 = fma(r3943255, r3943255, r3943260);
        double r3943267 = -1.0;
        double r3943268 = fma(r3943266, r3943266, r3943267);
        double r3943269 = fma(r3943264, r3943265, r3943268);
        return r3943269;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 +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))