Average Error: 17.1 → 0.5
Time: 3.1s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1 \]
↓
\[\mathsf{fma}\left(12, b \cdot b, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1 \]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(12, b \cdot b, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1

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

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 17.1

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

  2. Simplified15.9

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

  3. Taylor expanded in a around 0 0.5

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

  4. Simplified0.5

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

  5. Applied fma-udef_binary640.5

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

  6. Applied associate-+r+_binary640.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(12, b \cdot b, {\left(\mathsf{hypot}\left(a, b\right)\right)}^{4}\right) + -1 \]

Reproduce

herbie shell --seed 2022067 
(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))