Average Error: 0.2 → 0.2
Time: 25.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, b \cdot b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(4, b \cdot b, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r7500826 = a;
        double r7500827 = r7500826 * r7500826;
        double r7500828 = b;
        double r7500829 = r7500828 * r7500828;
        double r7500830 = r7500827 + r7500829;
        double r7500831 = 2.0;
        double r7500832 = pow(r7500830, r7500831);
        double r7500833 = 4.0;
        double r7500834 = r7500833 * r7500829;
        double r7500835 = r7500832 + r7500834;
        double r7500836 = 1.0;
        double r7500837 = r7500835 - r7500836;
        return r7500837;
}
↓
double f(double a, double b) {
        double r7500838 = 4.0;
        double r7500839 = b;
        double r7500840 = r7500839 * r7500839;
        double r7500841 = a;
        double r7500842 = fma(r7500841, r7500841, r7500840);
        double r7500843 = 2.0;
        double r7500844 = pow(r7500842, r7500843);
        double r7500845 = fma(r7500838, r7500840, r7500844);
        double r7500846 = 1.0;
        double r7500847 = r7500845 - r7500846;
        return r7500847;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019169 +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))