Average Error: 0.2 → 0.0
Time: 4.6s
Precision: binary64
Cost: 14272
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[-1 + \left(\left(b \cdot b\right) \cdot 4 + \left({a}^{4} + \left({b}^{4} + \left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\right)\right)\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
-1 + \left(\left(b \cdot b\right) \cdot 4 + \left({a}^{4} + \left({b}^{4} + \left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\right)\right)
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (+
  -1.0
  (+
   (* (* b b) 4.0)
   (+ (pow a 4.0) (+ (pow b 4.0) (* (* a a) (* b (* b 2.0))))))))
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 + (((b * b) * 4.0) + (pow(a, 4.0) + (pow(b, 4.0) + ((a * a) * (b * (b * 2.0))))));
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.2
Cost1472
\[-1 + \left(\left(b \cdot b\right) \cdot 4 + \left(b \cdot b + a \cdot a\right) \cdot \left(b \cdot b + a \cdot a\right)\right)\]
Alternative 2
Error11.3
Cost1088
\[-1 + b \cdot \left(b \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\right)\]
Alternative 3
Error11.8
Cost1288
\[\begin{array}{l} \mathbf{if}\;b \leq -0.947910129905006 \lor \neg \left(b \leq 0.9449220269269785\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(4 + \left(b \cdot b + \left(a \cdot a\right) \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\\ \end{array}\]
Alternative 4
Error22.2
Cost832
\[-1 + \left(b \cdot b\right) \cdot \left(4 + \left(a \cdot a\right) \cdot 2\right)\]
Alternative 5
Error23.0
Cost704
\[-1 + \left(b \cdot b\right) \cdot \left(\left(a \cdot a\right) \cdot 2\right)\]
Alternative 6
Error23.3
Cost64
\[-1\]
Alternative 7
Error62.2
Cost64
\[1\]

Error

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 0 0.0

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

  3. Simplified0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2021044 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))