Average Error: 0.2 → 0.9
Time: 4.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) + {a}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) + {a}^{4}\right) - 1
double code(double a, double b) {
	return ((pow(((a * a) + (b * b)), 2.0) + (4.0 * (b * b))) - 1.0);
}
↓
double code(double a, double b) {
	return ((fma((2.0 * pow(a, 2.0)), pow(b, 2.0), pow(b, 4.0)) + pow(a, 4.0)) - 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

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

  2. Taylor expanded around inf 0.9

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

  3. Simplified0.9

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))