Average Error: 0.2 → 0.2
Time: 4.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4 \cdot b, b, {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)\]
\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(a \cdot a + b \cdot b\right)}^{2} - 1\right)
double f(double a, double b) {
        double r212893 = a;
        double r212894 = r212893 * r212893;
        double r212895 = b;
        double r212896 = r212895 * r212895;
        double r212897 = r212894 + r212896;
        double r212898 = 2.0;
        double r212899 = pow(r212897, r212898);
        double r212900 = 4.0;
        double r212901 = r212900 * r212896;
        double r212902 = r212899 + r212901;
        double r212903 = 1.0;
        double r212904 = r212902 - r212903;
        return r212904;
}
↓
double f(double a, double b) {
        double r212905 = 4.0;
        double r212906 = b;
        double r212907 = r212905 * r212906;
        double r212908 = a;
        double r212909 = r212908 * r212908;
        double r212910 = r212906 * r212906;
        double r212911 = r212909 + r212910;
        double r212912 = 2.0;
        double r212913 = pow(r212911, r212912);
        double r212914 = 1.0;
        double r212915 = r212913 - r212914;
        double r212916 = fma(r212907, r212906, r212915);
        return r212916;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))