Bouland and Aaronson, Equation (26)

Average Error: 0.2 → 0.1

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot a \le 1.0752595847695641 \cdot 10^{-43}:\\ \;\;\;\;\left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\\ \end{array}\]

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) {
	double temp;
	if (((a * a) <= 1.075259584769564e-43)) {
		temp = (((pow(b, 4.0) + (2.0 * (pow(a, 2.0) * pow(b, 2.0)))) + (4.0 * (b * b))) - 1.0);
	} else {
		temp = ((pow(a, 4.0) + (pow(b, 4.0) + (2.0 * (pow(a, 2.0) * pow(b, 2.0))))) - 1.0);
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Derivation

1. Split input into 2 regimes

2. if (* a a) < 1.075259584769564e-43

   1. Initial program 0.1

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

   2. Taylor expanded around 0 0.0

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

   if 1.075259584769564e-43 < (* a a)

   1. Initial program 0.4

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

   2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} - 1\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \le 1.0752595847695641 \cdot 10^{-43}:\\ \;\;\;\;\left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))