Average Error: 0.2 → 0.0
Time: 6.1s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[\left(\left(4 \cdot {a}^{2} + \left(4 \cdot {a}^{3} + \left({a}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)\right)\right)\right) - 12 \cdot \left(a \cdot {b}^{2}\right)\right) + -1 \]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left(\left(4 \cdot {a}^{2} + \left(4 \cdot {a}^{3} + \left({a}^{4} + \left(2 \cdot \left({a}^{2} \cdot {b}^{2}\right) + \left(4 \cdot {b}^{2} + {b}^{4}\right)\right)\right)\right)\right) - 12 \cdot \left(a \cdot {b}^{2}\right)\right) + -1

(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
 (+
  (-
   (+
    (* 4.0 (pow a 2.0))
    (+
     (* 4.0 (pow a 3.0))
     (+
      (pow a 4.0)
      (+
       (* 2.0 (* (pow a 2.0) (pow b 2.0)))
       (+ (* 4.0 (pow b 2.0)) (pow b 4.0))))))
   (* 12.0 (* a (pow b 2.0))))
  -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) * (1.0 - (3.0 * a)))))) - 1.0;
}
↓
double code(double a, double b) {
	return (((4.0 * pow(a, 2.0)) + ((4.0 * pow(a, 3.0)) + (pow(a, 4.0) + ((2.0 * (pow(a, 2.0) * pow(b, 2.0))) + ((4.0 * pow(b, 2.0)) + pow(b, 4.0)))))) - (12.0 * (a * pow(b, 2.0)))) + -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

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. Simplified0.0

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

  3. Taylor expanded in b around 0 0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2021340 
(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))