Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 1.0

Time: 3.5s

Precision: 64

Math

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

↓

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

C

double code(double a, double b) {
	return ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0);
}

↓

double code(double a, double b) {
	return ((1.0 * (pow(b, 4.0) + (pow(a, 4.0) + (2.0 * (pow(a, 2.0) * pow(b, 2.0)))))) - 1.0);
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

2. Taylor expanded around inf 1.0

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

4. Applied *-un-lft-identity 1.0

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

5. Applied *-un-lft-identity 1.0

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

6. Applied distribute-lft-out 1.0

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

7. Simplified 1.0

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

8. Final simplification 1.0

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

Reproduce

herbie shell --seed 2020066 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))