Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 18.4s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r6205840 = a;
        double r6205841 = r6205840 * r6205840;
        double r6205842 = b;
        double r6205843 = r6205842 * r6205842;
        double r6205844 = r6205841 + r6205843;
        double r6205845 = 2.0;
        double r6205846 = pow(r6205844, r6205845);
        double r6205847 = 4.0;
        double r6205848 = 1.0;
        double r6205849 = r6205848 + r6205840;
        double r6205850 = r6205841 * r6205849;
        double r6205851 = 3.0;
        double r6205852 = r6205851 * r6205840;
        double r6205853 = r6205848 - r6205852;
        double r6205854 = r6205843 * r6205853;
        double r6205855 = r6205850 + r6205854;
        double r6205856 = r6205847 * r6205855;
        double r6205857 = r6205846 + r6205856;
        double r6205858 = r6205857 - r6205848;
        return r6205858;
}

↓

double f(double a, double b) {
        double r6205859 = 4.0;
        double r6205860 = a;
        double r6205861 = r6205860 * r6205860;
        double r6205862 = r6205861 * r6205860;
        double r6205863 = r6205859 * r6205862;
        double r6205864 = b;
        double r6205865 = r6205864 * r6205864;
        double r6205866 = r6205861 + r6205865;
        double r6205867 = -3.0;
        double r6205868 = r6205860 * r6205865;
        double r6205869 = r6205867 * r6205868;
        double r6205870 = r6205866 + r6205869;
        double r6205871 = r6205859 * r6205870;
        double r6205872 = -1.0;
        double r6205873 = sqrt(r6205866);
        double r6205874 = pow(r6205873, r6205859);
        double r6205875 = r6205872 + r6205874;
        double r6205876 = r6205871 + r6205875;
        double r6205877 = r6205863 + r6205876;
        return r6205877;
}

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. Simplified 0.2

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

4. Applied add-sqr-sqrt 0.2

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

5. Applied associate-*r* 0.1

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

7. Applied pow1 0.1

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

8. Applied add-sqr-sqrt 0.1

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

9. Applied pow3 0.1

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

10. Applied pow-prod-up 0.0

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

11. Simplified 0.0

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

12. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019162 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))