Average Error: 0.2 → 0.2
Time: 16.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\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 \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r351279 = a;
        double r351280 = r351279 * r351279;
        double r351281 = b;
        double r351282 = r351281 * r351281;
        double r351283 = r351280 + r351282;
        double r351284 = 2.0;
        double r351285 = pow(r351283, r351284);
        double r351286 = 4.0;
        double r351287 = 1.0;
        double r351288 = r351287 - r351279;
        double r351289 = r351280 * r351288;
        double r351290 = 3.0;
        double r351291 = r351290 + r351279;
        double r351292 = r351282 * r351291;
        double r351293 = r351289 + r351292;
        double r351294 = r351286 * r351293;
        double r351295 = r351285 + r351294;
        double r351296 = r351295 - r351287;
        return r351296;
}
↓
double f(double a, double b) {
        double r351297 = a;
        double r351298 = r351297 * r351297;
        double r351299 = 1.0;
        double r351300 = r351299 - r351297;
        double r351301 = b;
        double r351302 = r351301 * r351301;
        double r351303 = 3.0;
        double r351304 = r351303 + r351297;
        double r351305 = r351302 * r351304;
        double r351306 = fma(r351298, r351300, r351305);
        double r351307 = 4.0;
        double r351308 = fma(r351297, r351297, r351302);
        double r351309 = 2.0;
        double r351310 = pow(r351308, r351309);
        double r351311 = fma(r351306, r351307, r351310);
        double r351312 = r351311 - r351299;
        return r351312;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))