Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.1

Time: 34.3s

Precision: 64

Math

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

↓

\[\left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(\left(\left(b \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot 2 + b \cdot b\right) + {a}^{4}\right) + -12 \cdot \left(\left(b \cdot b\right) \cdot a\right)\right)\right) - 1\]

C

double f(double a, double b) {
        double r6157622 = a;
        double r6157623 = r6157622 * r6157622;
        double r6157624 = b;
        double r6157625 = r6157624 * r6157624;
        double r6157626 = r6157623 + r6157625;
        double r6157627 = 2.0;
        double r6157628 = pow(r6157626, r6157627);
        double r6157629 = 4.0;
        double r6157630 = 1.0;
        double r6157631 = r6157630 + r6157622;
        double r6157632 = r6157623 * r6157631;
        double r6157633 = 3.0;
        double r6157634 = r6157633 * r6157622;
        double r6157635 = r6157630 - r6157634;
        double r6157636 = r6157625 * r6157635;
        double r6157637 = r6157632 + r6157636;
        double r6157638 = r6157629 * r6157637;
        double r6157639 = r6157628 + r6157638;
        double r6157640 = r6157639 - r6157630;
        return r6157640;
}

↓

double f(double a, double b) {
        double r6157641 = a;
        double r6157642 = r6157641 * r6157641;
        double r6157643 = r6157642 * r6157641;
        double r6157644 = b;
        double r6157645 = r6157644 * r6157644;
        double r6157646 = r6157645 + r6157642;
        double r6157647 = r6157643 + r6157646;
        double r6157648 = 4.0;
        double r6157649 = r6157647 * r6157648;
        double r6157650 = 2.0;
        double r6157651 = r6157642 * r6157650;
        double r6157652 = r6157651 + r6157645;
        double r6157653 = r6157645 * r6157652;
        double r6157654 = pow(r6157641, r6157648);
        double r6157655 = r6157653 + r6157654;
        double r6157656 = -12.0;
        double r6157657 = r6157645 * r6157641;
        double r6157658 = r6157656 * r6157657;
        double r6157659 = r6157655 + r6157658;
        double r6157660 = r6157649 + r6157659;
        double r6157661 = 1.0;
        double r6157662 = r6157660 - r6157661;
        return r6157662;
}

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. Simplified 0.2

   \[\leadsto \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(a \cdot a + b \cdot b\right) \cdot \left(a \cdot a + b \cdot b\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1}\]

3. Taylor expanded around inf 0.0

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\color{blue}{\left({b}^{4} + \left({a}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

4. Simplified 0.2

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\color{blue}{\left(\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + \left(a \cdot a\right) \cdot \left(a \cdot a\right)\right)} + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]
5. Using strategy rm

6. Applied pow1 0.2

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + \left(a \cdot a\right) \cdot \left(\color{blue}{{a}^{1}} \cdot a\right)\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

7. Applied pow-plus 0.2

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + \left(a \cdot a\right) \cdot \color{blue}{{a}^{\left(1 + 1\right)}}\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

8. Applied pow1 0.2

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + \left(a \cdot \color{blue}{{a}^{1}}\right) \cdot {a}^{\left(1 + 1\right)}\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

9. Applied pow1 0.2

   \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + \left(\color{blue}{{a}^{1}} \cdot {a}^{1}\right) \cdot {a}^{\left(1 + 1\right)}\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

10. Applied pow-prod-up 0.2

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + \color{blue}{{a}^{\left(1 + 1\right)}} \cdot {a}^{\left(1 + 1\right)}\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

11. Applied pow-prod-up 0.1

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + \color{blue}{{a}^{\left(\left(1 + 1\right) + \left(1 + 1\right)\right)}}\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

12. Simplified 0.1

    \[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot a + \left(a \cdot a + b \cdot b\right)\right) + \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right) + {a}^{\color{blue}{4}}\right) + -12 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right) - 1\]

13. Final simplification 0.1

    \[\leadsto \left(\left(\left(a \cdot a\right) \cdot a + \left(b \cdot b + a \cdot a\right)\right) \cdot 4 + \left(\left(\left(b \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot 2 + b \cdot b\right) + {a}^{4}\right) + -12 \cdot \left(\left(b \cdot b\right) \cdot a\right)\right)\right) - 1\]

Reproduce

herbie shell --seed 2019142 
(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))