Average Error: 0.2 → 0.2
Time: 24.9s
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 r224014 = a;
        double r224015 = r224014 * r224014;
        double r224016 = b;
        double r224017 = r224016 * r224016;
        double r224018 = r224015 + r224017;
        double r224019 = 2.0;
        double r224020 = pow(r224018, r224019);
        double r224021 = 4.0;
        double r224022 = r224021 * r224017;
        double r224023 = r224020 + r224022;
        double r224024 = 1.0;
        double r224025 = r224023 - r224024;
        return r224025;
}
↓
double f(double a, double b) {
        double r224026 = 4.0;
        double r224027 = b;
        double r224028 = r224026 * r224027;
        double r224029 = a;
        double r224030 = r224027 * r224027;
        double r224031 = fma(r224029, r224029, r224030);
        double r224032 = 2.0;
        double r224033 = pow(r224031, r224032);
        double r224034 = fma(r224028, r224027, r224033);
        double r224035 = 1.0;
        double r224036 = r224034 - r224035;
        return r224036;
}

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 2019306 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))