Average Error: 0.2 → 0.2
Time: 4.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}, \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}, -1\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\mathsf{fma}\left(\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}, \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)}, -1\right)
double f(double a, double b) {
        double r236846 = a;
        double r236847 = r236846 * r236846;
        double r236848 = b;
        double r236849 = r236848 * r236848;
        double r236850 = r236847 + r236849;
        double r236851 = 2.0;
        double r236852 = pow(r236850, r236851);
        double r236853 = 4.0;
        double r236854 = r236853 * r236849;
        double r236855 = r236852 + r236854;
        double r236856 = 1.0;
        double r236857 = r236855 - r236856;
        return r236857;
}
↓
double f(double a, double b) {
        double r236858 = a;
        double r236859 = r236858 * r236858;
        double r236860 = b;
        double r236861 = r236860 * r236860;
        double r236862 = r236859 + r236861;
        double r236863 = 2.0;
        double r236864 = pow(r236862, r236863);
        double r236865 = 4.0;
        double r236866 = r236865 * r236861;
        double r236867 = r236864 + r236866;
        double r236868 = sqrt(r236867);
        double r236869 = 1.0;
        double r236870 = -r236869;
        double r236871 = fma(r236868, r236868, r236870);
        return r236871;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

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

  4. Applied fma-neg0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))