Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.1

Time: 15.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r6271507 = a;
        double r6271508 = r6271507 * r6271507;
        double r6271509 = b;
        double r6271510 = r6271509 * r6271509;
        double r6271511 = r6271508 + r6271510;
        double r6271512 = 2.0;
        double r6271513 = pow(r6271511, r6271512);
        double r6271514 = 4.0;
        double r6271515 = 1.0;
        double r6271516 = r6271515 - r6271507;
        double r6271517 = r6271508 * r6271516;
        double r6271518 = 3.0;
        double r6271519 = r6271518 + r6271507;
        double r6271520 = r6271510 * r6271519;
        double r6271521 = r6271517 + r6271520;
        double r6271522 = r6271514 * r6271521;
        double r6271523 = r6271513 + r6271522;
        double r6271524 = r6271523 - r6271515;
        return r6271524;
}

↓

double f(double a, double b) {
        double r6271525 = 4.0;
        double r6271526 = a;
        double r6271527 = r6271526 * r6271526;
        double r6271528 = 1.0;
        double r6271529 = r6271528 - r6271526;
        double r6271530 = r6271527 * r6271529;
        double r6271531 = 3.0;
        double r6271532 = r6271531 + r6271526;
        double r6271533 = b;
        double r6271534 = r6271533 * r6271533;
        double r6271535 = r6271532 * r6271534;
        double r6271536 = r6271530 + r6271535;
        double r6271537 = r6271525 * r6271536;
        double r6271538 = pow(r6271526, r6271525);
        double r6271539 = 2.0;
        double r6271540 = r6271527 * r6271539;
        double r6271541 = r6271534 + r6271540;
        double r6271542 = r6271533 * r6271541;
        double r6271543 = r6271542 * r6271533;
        double r6271544 = r6271538 + r6271543;
        double r6271545 = r6271537 + r6271544;
        double r6271546 = r6271545 - r6271528;
        return r6271546;
}

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

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

5. Applied pow1 0.2

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

6. Applied pow1 0.2

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

7. Applied pow-prod-up 0.2

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

8. Applied pow1 0.2

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

9. Applied pow1 0.2

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

10. Applied pow-prod-up 0.2

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

11. Applied pow-prod-up 0.1

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

12. Simplified 0.1

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

14. Applied associate-*l* 0.1

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

15. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019153 
(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))