Average Error: 0.2 → 0.2
Time: 5.0s
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 r361775 = a;
        double r361776 = r361775 * r361775;
        double r361777 = b;
        double r361778 = r361777 * r361777;
        double r361779 = r361776 + r361778;
        double r361780 = 2.0;
        double r361781 = pow(r361779, r361780);
        double r361782 = 4.0;
        double r361783 = r361782 * r361778;
        double r361784 = r361781 + r361783;
        double r361785 = 1.0;
        double r361786 = r361784 - r361785;
        return r361786;
}
↓
double f(double a, double b) {
        double r361787 = 4.0;
        double r361788 = b;
        double r361789 = r361787 * r361788;
        double r361790 = a;
        double r361791 = r361790 * r361790;
        double r361792 = r361788 * r361788;
        double r361793 = r361791 + r361792;
        double r361794 = 2.0;
        double r361795 = pow(r361793, r361794);
        double r361796 = 1.0;
        double r361797 = r361795 - r361796;
        double r361798 = fma(r361789, r361788, r361797);
        return r361798;
}

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))