Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.5

Time: 23.2s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r14656892 = a;
        double r14656893 = r14656892 * r14656892;
        double r14656894 = b;
        double r14656895 = r14656894 * r14656894;
        double r14656896 = r14656893 + r14656895;
        double r14656897 = 2.0;
        double r14656898 = pow(r14656896, r14656897);
        double r14656899 = 4.0;
        double r14656900 = 1.0;
        double r14656901 = r14656900 - r14656892;
        double r14656902 = r14656893 * r14656901;
        double r14656903 = 3.0;
        double r14656904 = r14656903 + r14656892;
        double r14656905 = r14656895 * r14656904;
        double r14656906 = r14656902 + r14656905;
        double r14656907 = r14656899 * r14656906;
        double r14656908 = r14656898 + r14656907;
        double r14656909 = r14656908 - r14656900;
        return r14656909;
}

↓

double f(double a, double b) {
        double r14656910 = a;
        double r14656911 = r14656910 * r14656910;
        double r14656912 = b;
        double r14656913 = r14656912 * r14656912;
        double r14656914 = r14656911 + r14656913;
        double r14656915 = 2.0;
        double r14656916 = pow(r14656914, r14656915);
        double r14656917 = cbrt(r14656916);
        double r14656918 = sqrt(r14656916);
        double r14656919 = cbrt(r14656918);
        double r14656920 = fabs(r14656918);
        double r14656921 = cbrt(r14656920);
        double r14656922 = r14656919 * r14656921;
        double r14656923 = r14656917 * r14656922;
        double r14656924 = r14656923 * r14656917;
        double r14656925 = 4.0;
        double r14656926 = 1.0;
        double r14656927 = r14656926 - r14656910;
        double r14656928 = r14656911 * r14656927;
        double r14656929 = 3.0;
        double r14656930 = r14656929 + r14656910;
        double r14656931 = r14656913 * r14656930;
        double r14656932 = r14656928 + r14656931;
        double r14656933 = r14656925 * r14656932;
        double r14656934 = r14656924 + r14656933;
        double r14656935 = r14656934 - r14656926;
        return r14656935;
}

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. Using strategy rm

3. Applied add-cube-cbrt 0.5

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

5. Applied add-sqr-sqrt 0.5

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

6. Applied cbrt-prod 0.5

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

8. Applied add-sqr-sqrt 0.5

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

9. Applied rem-sqrt-square 0.5

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

10. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019174 
(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))