Average Error: 0.2 → 0.5
Time: 3.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.0188075050240837 \lor \neg \left(a \le 0.99722586735940388\right):\\ \;\;\;\;\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4} + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, \mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) - 1\right)\\ \end{array}\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.0188075050240837 \lor \neg \left(a \le 0.99722586735940388\right):\\
\;\;\;\;\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4} + {a}^{4}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(4 \cdot b, b, \mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) - 1\right)\\

\end{array}
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) {
	double VAR;
	if (((a <= -1.0188075050240837) || !(a <= 0.9972258673594039))) {
		VAR = fma((2.0 * pow(a, 2.0)), pow(b, 2.0), (pow(b, 4.0) + pow(a, 4.0)));
	} else {
		VAR = fma((4.0 * b), b, (fma((2.0 * pow(a, 2.0)), pow(b, 2.0), pow(b, 4.0)) - 1.0));
	}
	return VAR;
}

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. Split input into 2 regimes

  2. if a < -1.0188075050240837 or 0.9972258673594039 < a

    1. Initial program 0.5

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

    2. Simplified0.5

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

    3. Taylor expanded around inf 1.0

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

    4. Simplified1.0

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

    if -1.0188075050240837 < a < 0.9972258673594039

    1. Initial program 0.1

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

    2. Simplified0.1

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

    3. Taylor expanded around 0 0.3

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

    4. Simplified0.3

      \[\leadsto \mathsf{fma}\left(4 \cdot b, b, \color{blue}{\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right)} - 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.0188075050240837 \lor \neg \left(a \le 0.99722586735940388\right):\\ \;\;\;\;\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4} + {a}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4 \cdot b, b, \mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))