Average Error: 0.2 → 0.0
Time: 4.9s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\left(4 \cdot {a}^{2} + \left(4 \cdot \left(a \cdot {b}^{2}\right) + \left({a}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({b}^{2} \cdot 12 + {b}^{4}\right)\right)\right)\right)\right) - \left(1 + 4 \cdot {a}^{3}\right) \]
\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 {a}^{2} + \left(4 \cdot \left(a \cdot {b}^{2}\right) + \left({a}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left({b}^{2} \cdot 12 + {b}^{4}\right)\right)\right)\right)\right) - \left(1 + 4 \cdot {a}^{3}\right)

(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 (pow a 2.0))
   (+
    (* 4.0 (* a (pow b 2.0)))
    (+
     (pow a 4.0)
     (+
      (* 2.0 (* (pow a 2.0) (pow b 2.0)))
      (+ (* (pow b 2.0) 12.0) (pow b 4.0))))))
  (+ 1.0 (* 4.0 (pow a 3.0)))))
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 * pow(a, 2.0)) + ((4.0 * (a * pow(b, 2.0))) + (pow(a, 4.0) + ((2.0 * (pow(a, 2.0) * pow(b, 2.0))) + ((pow(b, 2.0) * 12.0) + pow(b, 4.0)))))) - (1.0 + (4.0 * pow(a, 3.0)));
}

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

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

  2. Simplified0.0

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

  3. Taylor expanded in b around 0 0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022077 
(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))