Average Error: 0.2 → 1.0
Time: 4.7s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1
double f(double a, double b) {
        double r282362 = a;
        double r282363 = r282362 * r282362;
        double r282364 = b;
        double r282365 = r282364 * r282364;
        double r282366 = r282363 + r282365;
        double r282367 = 2.0;
        double r282368 = pow(r282366, r282367);
        double r282369 = 4.0;
        double r282370 = r282369 * r282365;
        double r282371 = r282368 + r282370;
        double r282372 = 1.0;
        double r282373 = r282371 - r282372;
        return r282373;
}
↓
double f(double a, double b) {
        double r282374 = a;
        double r282375 = 4.0;
        double r282376 = pow(r282374, r282375);
        double r282377 = b;
        double r282378 = pow(r282377, r282375);
        double r282379 = 2.0;
        double r282380 = pow(r282374, r282379);
        double r282381 = pow(r282377, r282379);
        double r282382 = r282380 * r282381;
        double r282383 = r282379 * r282382;
        double r282384 = r282378 + r282383;
        double r282385 = r282376 + r282384;
        double r282386 = 1.0;
        double r282387 = r282385 - r282386;
        return r282387;
}

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 1.0

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

  3. Final simplification1.0

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

Reproduce

herbie shell --seed 2020081 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))