Average Error: 0.2 → 0.2
Time: 5.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)\]
\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(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)
double f(double a, double b) {
        double r311859 = a;
        double r311860 = r311859 * r311859;
        double r311861 = b;
        double r311862 = r311861 * r311861;
        double r311863 = r311860 + r311862;
        double r311864 = 2.0;
        double r311865 = pow(r311863, r311864);
        double r311866 = 4.0;
        double r311867 = 1.0;
        double r311868 = r311867 - r311859;
        double r311869 = r311860 * r311868;
        double r311870 = 3.0;
        double r311871 = r311870 + r311859;
        double r311872 = r311862 * r311871;
        double r311873 = r311869 + r311872;
        double r311874 = r311866 * r311873;
        double r311875 = r311865 + r311874;
        double r311876 = r311875 - r311867;
        return r311876;
}
↓
double f(double a, double b) {
        double r311877 = 4.0;
        double r311878 = a;
        double r311879 = r311878 * r311878;
        double r311880 = 1.0;
        double r311881 = r311880 - r311878;
        double r311882 = b;
        double r311883 = r311882 * r311882;
        double r311884 = 3.0;
        double r311885 = r311884 + r311878;
        double r311886 = r311883 * r311885;
        double r311887 = fma(r311879, r311881, r311886);
        double r311888 = r311879 + r311883;
        double r311889 = 2.0;
        double r311890 = pow(r311888, r311889);
        double r311891 = r311890 - r311880;
        double r311892 = fma(r311877, r311887, r311891);
        return r311892;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))