Average Error: 0.2 → 0.2
Time: 29.2s
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 r179239 = a;
        double r179240 = r179239 * r179239;
        double r179241 = b;
        double r179242 = r179241 * r179241;
        double r179243 = r179240 + r179242;
        double r179244 = 2.0;
        double r179245 = pow(r179243, r179244);
        double r179246 = 4.0;
        double r179247 = r179246 * r179242;
        double r179248 = r179245 + r179247;
        double r179249 = 1.0;
        double r179250 = r179248 - r179249;
        return r179250;
}
↓
double f(double a, double b) {
        double r179251 = 4.0;
        double r179252 = b;
        double r179253 = r179251 * r179252;
        double r179254 = a;
        double r179255 = r179252 * r179252;
        double r179256 = fma(r179254, r179254, r179255);
        double r179257 = 2.0;
        double r179258 = pow(r179256, r179257);
        double r179259 = fma(r179253, r179252, r179258);
        double r179260 = 1.0;
        double r179261 = r179259 - r179260;
        return r179261;
}

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 2019323 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))