Average Error: 0.2 → 0.1
Time: 4.4s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[\left({a}^{4} + -1\right) + \left(a \cdot \left(a \cdot \left(2 \cdot \left(b \cdot b\right) + \left(4 + a \cdot 4\right)\right)\right) + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\right) \]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({a}^{4} + -1\right) + \left(a \cdot \left(a \cdot \left(2 \cdot \left(b \cdot b\right) + \left(4 + a \cdot 4\right)\right)\right) + \left({b}^{4} + 4 \cdot \left(b \cdot b\right)\right)\right)

(FPCore (a b)
 :precision binary64
 (-
  (+
   (pow (+ (* a a) (* b b)) 2.0)
   (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a))))))
  1.0))
↓
(FPCore (a b)
 :precision binary64
 (+
  (+ (pow a 4.0) -1.0)
  (+
   (* a (* a (+ (* 2.0 (* b b)) (+ 4.0 (* a 4.0)))))
   (+ (pow b 4.0) (* 4.0 (* b b))))))
double code(double a, double b) {
	return (pow(((a * a) + (b * b)), 2.0) + (4.0 * (((a * a) * (1.0 + a)) + ((b * b) * (1.0 - (3.0 * a)))))) - 1.0;
}
↓
double code(double a, double b) {
	return (pow(a, 4.0) + -1.0) + ((a * (a * ((2.0 * (b * b)) + (4.0 + (a * 4.0))))) + (pow(b, 4.0) + (4.0 * (b * b))));
}

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

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

  2. Taylor expanded in a around 0 0.0

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

  3. Simplified0.0

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

  4. Taylor expanded in a around 0 0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2021197 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))