Average Error: 0.2 → 0.0
Time: 2.2s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\left({b}^{4} + \left({a}^{4} + \left(a \cdot a\right) \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\left({b}^{4} + \left({a}^{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
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (-
  (+
   (+ (pow b 4.0) (+ (pow a 4.0) (* (* a a) (* b (* b 2.0)))))
   (* 4.0 (* b b)))
  1.0))
double code(double a, double b) {
	return ((double) (((double) (((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)) + ((double) (4.0 * ((double) (b * b)))))) - 1.0));
}
↓
double code(double a, double b) {
	return ((double) (((double) (((double) (((double) pow(b, 4.0)) + ((double) (((double) pow(a, 4.0)) + ((double) (((double) (a * a)) * ((double) (b * ((double) (b * 2.0)))))))))) + ((double) (4.0 * ((double) (b * b)))))) - 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

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020219 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))