Average Error: 0.2 → 0.2
Time: 2.8s
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 r232782 = a;
        double r232783 = r232782 * r232782;
        double r232784 = b;
        double r232785 = r232784 * r232784;
        double r232786 = r232783 + r232785;
        double r232787 = 2.0;
        double r232788 = pow(r232786, r232787);
        double r232789 = 4.0;
        double r232790 = r232789 * r232785;
        double r232791 = r232788 + r232790;
        double r232792 = 1.0;
        double r232793 = r232791 - r232792;
        return r232793;
}
↓
double f(double a, double b) {
        double r232794 = 4.0;
        double r232795 = b;
        double r232796 = r232794 * r232795;
        double r232797 = a;
        double r232798 = r232797 * r232797;
        double r232799 = r232795 * r232795;
        double r232800 = r232798 + r232799;
        double r232801 = 2.0;
        double r232802 = pow(r232800, r232801);
        double r232803 = 1.0;
        double r232804 = r232802 - r232803;
        double r232805 = fma(r232796, r232795, r232804);
        return r232805;
}

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 2020062 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))