Average Error: 0.2 → 0.2
Time: 26.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(a \cdot a + b \cdot b, a \cdot a + b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(a \cdot a + b \cdot b, a \cdot a + b \cdot b, \left(b \cdot b\right) \cdot 4\right) - 1
double f(double a, double b) {
        double r5526179 = a;
        double r5526180 = r5526179 * r5526179;
        double r5526181 = b;
        double r5526182 = r5526181 * r5526181;
        double r5526183 = r5526180 + r5526182;
        double r5526184 = 2.0;
        double r5526185 = pow(r5526183, r5526184);
        double r5526186 = 4.0;
        double r5526187 = r5526186 * r5526182;
        double r5526188 = r5526185 + r5526187;
        double r5526189 = 1.0;
        double r5526190 = r5526188 - r5526189;
        return r5526190;
}
↓
double f(double a, double b) {
        double r5526191 = a;
        double r5526192 = r5526191 * r5526191;
        double r5526193 = b;
        double r5526194 = r5526193 * r5526193;
        double r5526195 = r5526192 + r5526194;
        double r5526196 = 4.0;
        double r5526197 = r5526194 * r5526196;
        double r5526198 = fma(r5526195, r5526195, r5526197);
        double r5526199 = 1.0;
        double r5526200 = r5526198 - r5526199;
        return r5526200;
}

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. Using strategy rm

  3. Applied unpow20.2

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

  4. Applied fma-def0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))