Average Error: 0.2 → 0.2
Time: 3.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -30.610781749505392 \lor \neg \left(a \le 4.1950544641832567 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, 2 \cdot {a}^{2}\right), {a}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array}\]
\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 -30.610781749505392 \lor \neg \left(a \le 4.1950544641832567 \cdot 10^{-7}\right):\\
\;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, 2 \cdot {a}^{2}\right), {a}^{4}\right) - 1\\

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

\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 temp;
	if (((a <= -30.610781749505392) || !(a <= 4.1950544641832567e-07))) {
		temp = (fma(b, (b * fma(b, b, (2.0 * pow(a, 2.0)))), pow(a, 4.0)) - 1.0);
	} else {
		temp = ((fma((2.0 * pow(a, 2.0)), pow(b, 2.0), pow(b, 4.0)) + (4.0 * (b * b))) - 1.0);
	}
	return temp;
}

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 < -30.610781749505392 or 4.1950544641832567e-07 < 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. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -30.610781749505392 < a < 4.1950544641832567e-07

    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. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -30.610781749505392 \lor \neg \left(a \le 4.1950544641832567 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, b, 2 \cdot {a}^{2}\right), {a}^{4}\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(2 \cdot {a}^{2}, {b}^{2}, {b}^{4}\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))