Average Error: 0.2 → 0.2
Time: 1.1m
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(b \cdot 4, b, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(b \cdot 4, b, {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r8519844 = a;
        double r8519845 = r8519844 * r8519844;
        double r8519846 = b;
        double r8519847 = r8519846 * r8519846;
        double r8519848 = r8519845 + r8519847;
        double r8519849 = 2.0;
        double r8519850 = pow(r8519848, r8519849);
        double r8519851 = 4.0;
        double r8519852 = r8519851 * r8519847;
        double r8519853 = r8519850 + r8519852;
        double r8519854 = 1.0;
        double r8519855 = r8519853 - r8519854;
        return r8519855;
}
↓
double f(double a, double b) {
        double r8519856 = b;
        double r8519857 = 4.0;
        double r8519858 = r8519856 * r8519857;
        double r8519859 = a;
        double r8519860 = r8519859 * r8519859;
        double r8519861 = fma(r8519856, r8519856, r8519860);
        double r8519862 = 2.0;
        double r8519863 = pow(r8519861, r8519862);
        double r8519864 = fma(r8519858, r8519856, r8519863);
        double r8519865 = 1.0;
        double r8519866 = r8519864 - r8519865;
        return r8519866;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))