Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.0

Time: 16.9s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r2296441 = a;
        double r2296442 = r2296441 * r2296441;
        double r2296443 = b;
        double r2296444 = r2296443 * r2296443;
        double r2296445 = r2296442 + r2296444;
        double r2296446 = 2.0;
        double r2296447 = pow(r2296445, r2296446);
        double r2296448 = 4.0;
        double r2296449 = 1.0;
        double r2296450 = r2296449 + r2296441;
        double r2296451 = r2296442 * r2296450;
        double r2296452 = 3.0;
        double r2296453 = r2296452 * r2296441;
        double r2296454 = r2296449 - r2296453;
        double r2296455 = r2296444 * r2296454;
        double r2296456 = r2296451 + r2296455;
        double r2296457 = r2296448 * r2296456;
        double r2296458 = r2296447 + r2296457;
        double r2296459 = r2296458 - r2296449;
        return r2296459;
}

↓

double f(double a, double b) {
        double r2296460 = 1.0;
        double r2296461 = 3.0;
        double r2296462 = a;
        double r2296463 = r2296461 * r2296462;
        double r2296464 = r2296460 - r2296463;
        double r2296465 = b;
        double r2296466 = r2296465 * r2296465;
        double r2296467 = r2296464 * r2296466;
        double r2296468 = r2296462 + r2296460;
        double r2296469 = r2296462 * r2296462;
        double r2296470 = r2296468 * r2296469;
        double r2296471 = r2296467 + r2296470;
        double r2296472 = 4.0;
        double r2296473 = r2296471 * r2296472;
        double r2296474 = pow(r2296462, r2296472);
        double r2296475 = r2296466 * r2296469;
        double r2296476 = 2.0;
        double r2296477 = r2296475 * r2296476;
        double r2296478 = r2296474 + r2296477;
        double r2296479 = pow(r2296465, r2296472);
        double r2296480 = r2296478 + r2296479;
        double r2296481 = r2296473 + r2296480;
        double r2296482 = r2296481 - r2296460;
        return r2296482;
}

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. Taylor expanded around 0 0.0

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

3. Simplified 0.2

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

5. Applied pow1 0.2

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

6. Applied pow1 0.2

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

7. Applied pow-prod-up 0.2

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

8. Applied pow1 0.2

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

9. Applied pow1 0.2

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

10. Applied pow-prod-up 0.2

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

11. Applied pow-prod-up 0.1

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

12. Simplified 0.1

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

14. Applied pow1 0.1

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

15. Applied pow1 0.1

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

16. Applied pow-prod-up 0.1

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

17. Applied pow1 0.1

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

18. Applied pow1 0.1

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

19. Applied pow-prod-up 0.1

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

20. Applied pow-prod-up 0.0

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

21. Simplified 0.0

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

22. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019153 
(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))