Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.1

Time: 22.3s

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 r8147711 = a;
        double r8147712 = r8147711 * r8147711;
        double r8147713 = b;
        double r8147714 = r8147713 * r8147713;
        double r8147715 = r8147712 + r8147714;
        double r8147716 = 2.0;
        double r8147717 = pow(r8147715, r8147716);
        double r8147718 = 4.0;
        double r8147719 = 1.0;
        double r8147720 = r8147719 - r8147711;
        double r8147721 = r8147712 * r8147720;
        double r8147722 = 3.0;
        double r8147723 = r8147722 + r8147711;
        double r8147724 = r8147714 * r8147723;
        double r8147725 = r8147721 + r8147724;
        double r8147726 = r8147718 * r8147725;
        double r8147727 = r8147717 + r8147726;
        double r8147728 = r8147727 - r8147719;
        return r8147728;
}

↓

double f(double a, double b) {
        double r8147729 = 4.0;
        double r8147730 = a;
        double r8147731 = r8147730 * r8147730;
        double r8147732 = 1.0;
        double r8147733 = r8147732 - r8147730;
        double r8147734 = r8147731 * r8147733;
        double r8147735 = 3.0;
        double r8147736 = r8147730 + r8147735;
        double r8147737 = b;
        double r8147738 = r8147737 * r8147737;
        double r8147739 = r8147736 * r8147738;
        double r8147740 = r8147734 + r8147739;
        double r8147741 = r8147729 * r8147740;
        double r8147742 = pow(r8147730, r8147729);
        double r8147743 = 2.0;
        double r8147744 = r8147731 * r8147743;
        double r8147745 = r8147738 + r8147744;
        double r8147746 = r8147737 * r8147745;
        double r8147747 = r8147746 * r8147737;
        double r8147748 = r8147742 + r8147747;
        double r8147749 = r8147741 + r8147748;
        double r8147750 = r8147749 - r8147732;
        return r8147750;
}

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 -inf 0.0

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

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 pow1 0.2

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

6. Applied pow-plus 0.2

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

7. Applied pow1 0.2

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

8. Applied pow-plus 0.2

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

9. Applied pow-prod-up 0.1

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

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 2019130 
(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))