Average Error: 0.2 → 0.0

Time: 6.6s

Precision: binary64

Cost: 15040

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

↓

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

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

(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
 (-
  (+
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ a 3.0))))
   (+ (pow a 4.0) (+ (pow b 4.0) (* 2.0 (* (* a a) (* b b))))))
  1.0))

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 ((4.0 * (((a * a) * (1.0 - a)) + ((b * b) * (a + 3.0)))) + (pow(a, 4.0) + (pow(b, 4.0) + (2.0 * ((a * a) * (b * b)))))) - 1.0;
}

Error

The X axis uses an exponential scale — Bits error versus `a`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	0.2
Cost	14784

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

Alternative 2
Error	0.2
Cost	2240

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

Alternative 3
Error	1.6
Cost	2242

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6670250584883396 \cdot 10^{-06}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(4 + \left(b \cdot b\right) \cdot 2\right) + a \cdot \left(a - 4\right)\right) - 1\\ \mathbf{elif}\;a \leq 0.005744048479646246:\\ \;\;\;\;\left(4 \cdot \left(a \cdot a\right) + \left(b \cdot b\right) \cdot \left(a \cdot \left(4 + a \cdot 2\right) + \left(b \cdot b + 12\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot \left(4 + \left(b \cdot b\right) \cdot 2\right) + a \cdot \left(a \cdot \left(a \cdot \left(a - 4\right)\right)\right)\right) - 1\\ \end{array}\]

Alternative 4
Error	1.7
Cost	2114

\[\begin{array}{l} \mathbf{if}\;a \leq -4.922910256473494 \cdot 10^{-09}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(4 + \left(b \cdot b\right) \cdot 2\right) + a \cdot \left(a - 4\right)\right) - 1\\ \mathbf{elif}\;a \leq 0.012463999438901438:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(12 + 4 \cdot a\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot \left(4 + \left(b \cdot b\right) \cdot 2\right) + a \cdot \left(a \cdot \left(a \cdot \left(a - 4\right)\right)\right)\right) - 1\\ \end{array}\]

Alternative 5
Error	1.7
Cost	1544

\[\begin{array}{l} \mathbf{if}\;a \leq -1.614473810573396 \cdot 10^{-08} \lor \neg \left(a \leq 0.003177323774896446\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(\left(4 + \left(b \cdot b\right) \cdot 2\right) + a \cdot \left(a - 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(12 + 4 \cdot a\right)\right) - 1\\ \end{array}\]

Alternative 6
Error	1.8
Cost	1288

\[\begin{array}{l} \mathbf{if}\;a \leq -1.614473810573396 \cdot 10^{-08} \lor \neg \left(a \leq 0.002628048619920887\right):\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(4 + a \cdot \left(a - 4\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(b \cdot b + \left(12 + 4 \cdot a\right)\right) - 1\\ \end{array}\]

Alternative 7
Error	12.1
Cost	832

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

Alternative 8
Error	22.7
Cost	832

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

Alternative 9
Error	23.3
Cost	704

\[\left(a \cdot a\right) \cdot \left(\left(b \cdot b\right) \cdot 2\right) - 1\]

Alternative 10
Error	23.6
Cost	64

\[-1\]

Alternative 11
Error	62.2
Cost	64

\[1\]

Derivation

Initial program 0.2
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
Taylor expanded around 0 0.0
\[\leadsto \left(\color{blue}{\left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({b}^{4} + {a}^{4}\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
Simplified0.0
\[\leadsto \left(\color{blue}{\left({a}^{4} + \left({b}^{4} + 2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)\right)\right)} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
Simplified0.0
\[\leadsto \color{blue}{\left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + \left({a}^{4} + \left({b}^{4} + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) - 1}\]
Final simplification0.0
\[\leadsto \left(4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(a + 3\right)\right) + \left({a}^{4} + \left({b}^{4} + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) - 1\]

Reproduce

herbie shell --seed 2021044 
(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))