Average Error: 0.2 → 0.2
Time: 18.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, 3, \mathsf{fma}\left(1 - a, a, b \cdot b\right) \cdot a\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, 3, \mathsf{fma}\left(1 - a, a, b \cdot b\right) \cdot a\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)
double f(double a, double b) {
        double r8902877 = a;
        double r8902878 = r8902877 * r8902877;
        double r8902879 = b;
        double r8902880 = r8902879 * r8902879;
        double r8902881 = r8902878 + r8902880;
        double r8902882 = 2.0;
        double r8902883 = pow(r8902881, r8902882);
        double r8902884 = 4.0;
        double r8902885 = 1.0;
        double r8902886 = r8902885 - r8902877;
        double r8902887 = r8902878 * r8902886;
        double r8902888 = 3.0;
        double r8902889 = r8902888 + r8902877;
        double r8902890 = r8902880 * r8902889;
        double r8902891 = r8902887 + r8902890;
        double r8902892 = r8902884 * r8902891;
        double r8902893 = r8902883 + r8902892;
        double r8902894 = r8902893 - r8902885;
        return r8902894;
}
↓
double f(double a, double b) {
        double r8902895 = 4.0;
        double r8902896 = b;
        double r8902897 = r8902896 * r8902896;
        double r8902898 = 3.0;
        double r8902899 = 1.0;
        double r8902900 = a;
        double r8902901 = r8902899 - r8902900;
        double r8902902 = fma(r8902901, r8902900, r8902897);
        double r8902903 = r8902902 * r8902900;
        double r8902904 = fma(r8902897, r8902898, r8902903);
        double r8902905 = r8902900 * r8902900;
        double r8902906 = fma(r8902896, r8902896, r8902905);
        double r8902907 = 2.0;
        double r8902908 = pow(r8902906, r8902907);
        double r8902909 = r8902908 - r8902899;
        double r8902910 = fma(r8902895, r8902904, r8902909);
        return r8902910;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, 3, \mathsf{fma}\left(1 - a, a, b \cdot b\right) \cdot a\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)}\]

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(b \cdot b, 3, \mathsf{fma}\left(1 - a, a, b \cdot b\right) \cdot a\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2} - 1\right)\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))