Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.1

Time: 25.2s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r8305789 = a;
        double r8305790 = r8305789 * r8305789;
        double r8305791 = b;
        double r8305792 = r8305791 * r8305791;
        double r8305793 = r8305790 + r8305792;
        double r8305794 = 2.0;
        double r8305795 = pow(r8305793, r8305794);
        double r8305796 = 4.0;
        double r8305797 = 1.0;
        double r8305798 = r8305797 - r8305789;
        double r8305799 = r8305790 * r8305798;
        double r8305800 = 3.0;
        double r8305801 = r8305800 + r8305789;
        double r8305802 = r8305792 * r8305801;
        double r8305803 = r8305799 + r8305802;
        double r8305804 = r8305796 * r8305803;
        double r8305805 = r8305795 + r8305804;
        double r8305806 = r8305805 - r8305797;
        return r8305806;
}

↓

double f(double a, double b) {
        double r8305807 = 4.0;
        double r8305808 = a;
        double r8305809 = r8305808 * r8305808;
        double r8305810 = 1.0;
        double r8305811 = r8305810 - r8305808;
        double r8305812 = r8305809 * r8305811;
        double r8305813 = 3.0;
        double r8305814 = r8305808 + r8305813;
        double r8305815 = b;
        double r8305816 = r8305815 * r8305815;
        double r8305817 = r8305814 * r8305816;
        double r8305818 = r8305812 + r8305817;
        double r8305819 = r8305807 * r8305818;
        double r8305820 = pow(r8305808, r8305807);
        double r8305821 = 2.0;
        double r8305822 = r8305809 * r8305821;
        double r8305823 = r8305816 + r8305822;
        double r8305824 = r8305815 * r8305823;
        double r8305825 = r8305824 * r8305815;
        double r8305826 = r8305820 + r8305825;
        double r8305827 = r8305819 + r8305826;
        double r8305828 = r8305827 - r8305810;
        return r8305828;
}

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. Taylor expanded around 0 0.0

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

3. Simplified 0.2

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

5. Applied pow2 0.2

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

6. Applied pow1 0.2

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

7. Applied pow1 0.2

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

8. Applied pow-prod-up 0.2

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

9. Applied pow-prod-up 0.1

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

10. Simplified 0.1

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

12. Applied associate-*l* 0.1

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

13. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019142 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))