Average Error: 0.2 → 0.5
Time: 1.9s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -13027.803644546246 \lor \neg \left(a \leq 7.585443724790434 \cdot 10^{-102}\right):\\ \;\;\;\;\left({b}^{4} + \left({a}^{4} + a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left({b}^{\left(2 \cdot 2\right)} + a \cdot \left(a \cdot \left(2 \cdot {b}^{2}\right)\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array}\]
\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 \leq -13027.803644546246 \lor \neg \left(a \leq 7.585443724790434 \cdot 10^{-102}\right):\\
\;\;\;\;\left({b}^{4} + \left({a}^{4} + a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\right)\right) - 1\\

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

\end{array}
(FPCore (a b)
 :precision binary64
 (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))
↓
(FPCore (a b)
 :precision binary64
 (if (or (<= a -13027.803644546246) (not (<= a 7.585443724790434e-102)))
   (- (+ (pow b 4.0) (+ (pow a 4.0) (* a (* a (* b (* b 2.0)))))) 1.0)
   (-
    (+ (+ (pow b (* 2.0 2.0)) (* a (* a (* 2.0 (pow b 2.0))))) (* 4.0 (* b b)))
    1.0)))
double code(double a, double b) {
	return ((double) (((double) (((double) pow(((double) (((double) (a * a)) + ((double) (b * b)))), 2.0)) + ((double) (4.0 * ((double) (b * b)))))) - 1.0));
}
↓
double code(double a, double b) {
	double VAR;
	if (((a <= -13027.803644546246) || !(a <= 7.585443724790434e-102))) {
		VAR = ((double) (((double) (((double) pow(b, 4.0)) + ((double) (((double) pow(a, 4.0)) + ((double) (a * ((double) (a * ((double) (b * ((double) (b * 2.0)))))))))))) - 1.0));
	} else {
		VAR = ((double) (((double) (((double) (((double) pow(b, ((double) (2.0 * 2.0)))) + ((double) (a * ((double) (a * ((double) (2.0 * ((double) pow(b, 2.0)))))))))) + ((double) (4.0 * ((double) (b * b)))))) - 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 < -13027.803644546246 or 7.58544372479043434e-102 < a

    1. Initial program 0.4

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.5

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

    if -13027.803644546246 < a < 7.58544372479043434e-102

    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 32.5

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

    3. Simplified0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -13027.803644546246 \lor \neg \left(a \leq 7.585443724790434 \cdot 10^{-102}\right):\\ \;\;\;\;\left({b}^{4} + \left({a}^{4} + a \cdot \left(a \cdot \left(b \cdot \left(b \cdot 2\right)\right)\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left({b}^{\left(2 \cdot 2\right)} + a \cdot \left(a \cdot \left(2 \cdot {b}^{2}\right)\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))