Average Error: 0.2 → 0.0
Time: 6.0s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1 \]
↓
\[\left(\mathsf{fma}\left(4, \mathsf{fma}\left(a, a, {a}^{3}\right), \mathsf{fma}\left(4, b \cdot b, {b}^{4}\right) + a \cdot \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(2, a, -12\right)\right)\right) + {a}^{4}\right) + -1 \]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left(\mathsf{fma}\left(4, \mathsf{fma}\left(a, a, {a}^{3}\right), \mathsf{fma}\left(4, b \cdot b, {b}^{4}\right) + a \cdot \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(2, a, -12\right)\right)\right) + {a}^{4}\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
 (+
  (+
   (fma
    4.0
    (fma a a (pow a 3.0))
    (+ (fma 4.0 (* b b) (pow b 4.0)) (* a (* (* b b) (fma 2.0 a -12.0)))))
   (pow a 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) * (1.0 - (3.0 * a)))))) - 1.0;
}
↓
double code(double a, double b) {
	return (fma(4.0, fma(a, a, pow(a, 3.0)), (fma(4.0, (b * b), pow(b, 4.0)) + (a * ((b * b) * fma(2.0, a, -12.0))))) + pow(a, 4.0)) + -1.0;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  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. Simplified0.1

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

  5. Applied distribute-rgt-in_binary640.1

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2021313 
(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))