Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.5

Time: 7.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r197861 = a;
        double r197862 = r197861 * r197861;
        double r197863 = b;
        double r197864 = r197863 * r197863;
        double r197865 = r197862 + r197864;
        double r197866 = 2.0;
        double r197867 = pow(r197865, r197866);
        double r197868 = 4.0;
        double r197869 = 1.0;
        double r197870 = r197869 + r197861;
        double r197871 = r197862 * r197870;
        double r197872 = 3.0;
        double r197873 = r197872 * r197861;
        double r197874 = r197869 - r197873;
        double r197875 = r197864 * r197874;
        double r197876 = r197871 + r197875;
        double r197877 = r197868 * r197876;
        double r197878 = r197867 + r197877;
        double r197879 = r197878 - r197869;
        return r197879;
}

↓

double f(double a, double b) {
        double r197880 = a;
        double r197881 = r197880 * r197880;
        double r197882 = b;
        double r197883 = r197882 * r197882;
        double r197884 = r197881 + r197883;
        double r197885 = sqrt(r197884);
        double r197886 = 2.0;
        double r197887 = pow(r197885, r197886);
        double r197888 = cbrt(r197887);
        double r197889 = pow(r197884, r197886);
        double r197890 = cbrt(r197889);
        double r197891 = r197888 * r197888;
        double r197892 = cbrt(r197891);
        double r197893 = cbrt(r197888);
        double r197894 = r197892 * r197893;
        double r197895 = r197890 * r197894;
        double r197896 = r197888 * r197895;
        double r197897 = r197896 * r197890;
        double r197898 = 4.0;
        double r197899 = 1.0;
        double r197900 = r197899 + r197880;
        double r197901 = r197881 * r197900;
        double r197902 = 3.0;
        double r197903 = r197902 * r197880;
        double r197904 = r197899 - r197903;
        double r197905 = r197883 * r197904;
        double r197906 = r197901 + r197905;
        double r197907 = r197898 * r197906;
        double r197908 = r197897 + r197907;
        double r197909 = r197908 - r197899;
        return r197909;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

3. Applied add-cube-cbrt 0.5

   \[\leadsto \left(\color{blue}{\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}\right) \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
4. Using strategy rm

5. Applied add-sqr-sqrt 0.5

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

6. Applied unpow-prod-down 0.5

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

7. Applied cbrt-prod 0.5

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

8. Applied associate-*l* 0.5

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

9. Simplified 0.5

   \[\leadsto \left(\left(\sqrt[3]{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt{a \cdot a + b \cdot b}\right)}^{2}}\right)}\right) \cdot \sqrt[3]{{\left(a \cdot a + b \cdot b\right)}^{2}} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
10. Using strategy rm

11. Applied add-cube-cbrt 0.5

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

12. Applied cbrt-prod 0.5

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

13. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019352 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))