Average Error: 0.2 → 0.0
Time: 1.9s
Precision: binary64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot a \leq 4.456330607025755 \cdot 10^{-21}:\\ \;\;\;\;\left(\left({b}^{4} + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + \left({b}^{4} + \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right)\right) - 1\\ \end{array}\]
\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 \cdot a \leq 4.456330607025755 \cdot 10^{-21}:\\
\;\;\;\;\left(\left({b}^{4} + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) + \left(b \cdot b\right) \cdot 4\right) - 1\\

\mathbf{else}:\\
\;\;\;\;\left({a}^{4} + \left({b}^{4} + \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right) \cdot 2\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 (<= (* a a) 4.456330607025755e-21)
   (- (+ (+ (pow b 4.0) (* 2.0 (* (* a a) (* b b)))) (* (* b b) 4.0)) 1.0)
   (- (+ (pow a 4.0) (+ (pow b 4.0) (* (* (* a a) (* b b)) 2.0))) 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 tmp;
	if ((((double) (a * a)) <= 4.456330607025755e-21)) {
		tmp = ((double) (((double) (((double) (((double) pow(b, 4.0)) + ((double) (2.0 * ((double) (((double) (a * a)) * ((double) (b * b)))))))) + ((double) (((double) (b * b)) * 4.0)))) - 1.0));
	} else {
		tmp = ((double) (((double) (((double) pow(a, 4.0)) + ((double) (((double) pow(b, 4.0)) + ((double) (((double) (((double) (a * a)) * ((double) (b * b)))) * 2.0)))))) - 1.0));
	}
	return tmp;
}

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 (*.f64 a a) < 4.4563306070257551e-21

    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 0.0

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

    3. Simplified0.0

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

    if 4.4563306070257551e-21 < (*.f64 a 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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 4.456330607025755 \cdot 10^{-21}:\\ \;\;\;\;\left(\left({b}^{4} + 2 \cdot \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) + \left(b \cdot b\right) \cdot 4\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + \left({b}^{4} + \left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right) \cdot 2\right)\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (* b b))) 1.0))