Average Error: 0.2 → 0.0
Time: 18.8s
Precision: binary64
Cost: 14912
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left(\left({a}^{4} + \left({b}^{4} + 2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)\right)\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot 3\right)\right) - 1\]
\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(\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 3\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
 (-
  (+
   (+ (pow a 4.0) (+ (pow b 4.0) (* 2.0 (* (* b b) (* a a)))))
   (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) 3.0))))
  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 ((pow(a, 4.0) + (pow(b, 4.0) + (2.0 * ((b * b) * (a * a))))) + (4.0 * (((a * a) * (1.0 - a)) + ((b * b) * 3.0)))) - 1.0;
}

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
Error2.4
Cost14272
\[\left(\left({a}^{4} + \left({b}^{4} + \left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]
Alternative 2
Error2.2
Cost14528
\[\left(\left({a}^{4} + \left({b}^{4} + 2 \cdot \left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)\right)\right) + 4 \cdot \left(b \cdot b + a \cdot a\right)\right) - 1\]

Error

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

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

  3. Simplified0.0

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2021043 
(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))