Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.2

Time: 4.2s

Precision: binary64

Math

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

↓

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

C

double code(double a, double b) {
	return ((double) (((double) (((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)) + ((double) (4.0 * ((double) (((double) (((double) (a * a)) * ((double) (1.0 - a)))) + ((double) (((double) (b * b)) * ((double) (3.0 + a)))))))))) - 1.0));
}

↓

double code(double a, double b) {
	return ((double) (((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)) + ((double) (((double) (4.0 * ((double) (((double) (a * ((double) (a * ((double) (1.0 - a)))))) + ((double) (b * ((double) (b * ((double) (a + 3.0)))))))))) - 1.0))));
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

3. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020184 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))