Bouland and Aaronson, Equation (24)

Average Error: 0.2 → 0.4

Time: 2.3min

Precision: binary64

Cost: 14272

Math

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

↓

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

FPCore

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a)))))
  1.0))

↓

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow b 4.0)
   (+ (* (* b b) 12.0) (+ (pow a 4.0) (* a (* 4.0 (- a (* a a)))))))
  1.0))

C

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

↓

double code(double a, double b) {
	return (pow(b, 4.0) + (((b * b) * 12.0) + (pow(a, 4.0) + (a * (4.0 * (a - (a * a))))))) - 1.0;
}

Error

Bits error versus a

Bits error versus b

Alternatives

Alternative 1
Error	0.2
Cost	2240

\[\left(\left(b \cdot b + a \cdot a\right) \cdot \left(b \cdot b + a \cdot a\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right)\right) - 1\]

Alternative 2
Error	2.3
Cost	2184

\[\begin{array}{l} \mathbf{if}\;a \leq -3.7422719384935035 \lor \neg \left(a \leq 3491160.435770424\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + -4\right) + \left(b \cdot b\right) \cdot 2\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(4 \cdot \left(a - a \cdot a\right)\right) + \left(b \cdot b\right) \cdot \left(a \cdot \left(4 + a \cdot 2\right) + \left(b \cdot b + 12\right)\right)\right) - 1\\ \end{array}\]

Alternative 3
Error	2.4
Cost	1928

\[\begin{array}{l} \mathbf{if}\;a \leq -3.7422719384935035 \lor \neg \left(a \leq 3491160.435770424\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + -4\right) + \left(b \cdot b\right) \cdot 2\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot \left(a \cdot \left(4 + a \cdot 2\right) + \left(b \cdot b + 12\right)\right) + a \cdot \left(4 \cdot a\right)\right) - 1\\ \end{array}\]

Alternative 4
Error	2.7
Cost	1416

\[\begin{array}{l} \mathbf{if}\;a \leq -1.611460634106778 \cdot 10^{-05} \lor \neg \left(a \leq 3931742.640037186\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(a \cdot \left(a + -4\right) + \left(b \cdot b\right) \cdot 2\right) - 1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(b \cdot b + \left(12 + 4 \cdot a\right)\right)\right) - 1\\ \end{array}\]

Alternative 5
Error	12.5
Cost	960

\[b \cdot \left(b \cdot \left(b \cdot b + \left(12 + 4 \cdot a\right)\right)\right) - 1\]

Alternative 6
Error	12.5
Cost	960

\[\left(b \cdot b\right) \cdot \left(b \cdot b + \left(12 + 4 \cdot a\right)\right) - 1\]

Alternative 7
Error	12.5
Cost	704

\[\left(b \cdot b\right) \cdot \left(b \cdot b + 12\right) - 1\]

Alternative 8
Error	13.5
Cost	576

\[\left(b \cdot b\right) \cdot \left(b \cdot b\right) - 1\]

Alternative 9
Error	24.0
Cost	64

\[-1\]

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

   \[\leadsto \color{blue}{\left(\left(12 \cdot {b}^{2} + \left({b}^{4} + \left(4 \cdot \left(a \cdot {b}^{2}\right) + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left(4 \cdot {a}^{2} + {a}^{4}\right)\right)\right)\right)\right) - 4 \cdot {a}^{3}\right)} - 1\]

3. Simplified 0.0

   \[\leadsto \color{blue}{\left({b}^{4} + \left(\left(b \cdot b\right) \cdot \left(12 + a \cdot \left(4 + 2 \cdot a\right)\right) + \left({a}^{4} + \left(a \cdot a - {a}^{3}\right) \cdot 4\right)\right)\right)} - 1\]

4. Taylor expanded around 0 0.4

   \[\leadsto \left({b}^{4} + \left(\color{blue}{12 \cdot {b}^{2}} + \left({a}^{4} + \left(a \cdot a - {a}^{3}\right) \cdot 4\right)\right)\right) - 1\]

5. Simplified 0.4

   \[\leadsto \left({b}^{4} + \left(\color{blue}{\left(b \cdot b\right) \cdot 12} + \left({a}^{4} + \left(a \cdot a - {a}^{3}\right) \cdot 4\right)\right)\right) - 1\]
6. Using strategy rm

7. Applied unpow3_binary64_3213 0.4

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

8. Applied distribute-rgt-out--_binary64_3101 0.4

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

9. Applied associate-*l*_binary64_3088 0.4

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

10. Simplified 0.4

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

11. Simplified 0.4

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

12. Final simplification 0.4

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

Reproduce

herbie shell --seed 2021042 
(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))