Average Error: 0.2 → 0.1
Time: 5.9s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[{\left(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 - a, 3 \cdot \left(b \cdot b\right)\right), -1\right) \]
\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(\mathsf{hypot}\left(a, b\right)\right)}^{4} + \mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 - a, 3 \cdot \left(b \cdot b\right)\right), -1\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
 (+
  (pow (hypot a b) 4.0)
  (fma 4.0 (fma (* a a) (- 1.0 a) (* 3.0 (* b b))) -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(hypot(a, b), 4.0) + fma(4.0, fma((a * a), (1.0 - a), (3.0 * (b * b))), -1.0);
}

Error

Bits error versus a

Bits error versus b

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 a around 0 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022104 
(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))