Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 31.8s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r11950917 = a;
        double r11950918 = r11950917 * r11950917;
        double r11950919 = b;
        double r11950920 = r11950919 * r11950919;
        double r11950921 = r11950918 + r11950920;
        double r11950922 = 2.0;
        double r11950923 = pow(r11950921, r11950922);
        double r11950924 = 4.0;
        double r11950925 = 1.0;
        double r11950926 = r11950925 + r11950917;
        double r11950927 = r11950918 * r11950926;
        double r11950928 = 3.0;
        double r11950929 = r11950928 * r11950917;
        double r11950930 = r11950925 - r11950929;
        double r11950931 = r11950920 * r11950930;
        double r11950932 = r11950927 + r11950931;
        double r11950933 = r11950924 * r11950932;
        double r11950934 = r11950923 + r11950933;
        double r11950935 = r11950934 - r11950925;
        return r11950935;
}

↓

double f(double a, double b) {
        double r11950936 = b;
        double r11950937 = r11950936 * r11950936;
        double r11950938 = a;
        double r11950939 = r11950938 * r11950938;
        double r11950940 = r11950938 * r11950939;
        double r11950941 = r11950939 + r11950940;
        double r11950942 = r11950937 + r11950941;
        double r11950943 = 4.0;
        double r11950944 = r11950942 * r11950943;
        double r11950945 = r11950939 + r11950937;
        double r11950946 = sqrt(r11950945);
        double r11950947 = pow(r11950946, r11950943);
        double r11950948 = r11950944 + r11950947;
        double r11950949 = -4.0;
        double r11950950 = 3.0;
        double r11950951 = r11950950 * r11950938;
        double r11950952 = r11950951 * r11950937;
        double r11950953 = r11950949 * r11950952;
        double r11950954 = r11950948 + r11950953;
        double r11950955 = 1.0;
        double r11950956 = r11950954 - r11950955;
        return r11950956;
}

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 sub-neg 0.2

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

4. Applied distribute-rgt-in 0.2

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

5. Applied associate-+r+ 0.2

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

6. Applied distribute-rgt-in 0.2

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

7. Applied associate-+r+ 0.2

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

8. Simplified 0.2

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

10. Applied add-sqr-sqrt 0.2

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

11. Applied associate-*r* 0.1

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

13. Applied add-sqr-sqrt 0.1

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

14. Applied pow3 0.1

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

16. Applied pow1 0.1

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

17. Applied pow-prod-up 0.0

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

18. Simplified 0.0

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

19. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019158 
(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))