Average Error: 0.2 → 1.0
Time: 2.2s
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 r189372 = a;
        double r189373 = r189372 * r189372;
        double r189374 = b;
        double r189375 = r189374 * r189374;
        double r189376 = r189373 + r189375;
        double r189377 = 2.0;
        double r189378 = pow(r189376, r189377);
        double r189379 = 4.0;
        double r189380 = r189379 * r189375;
        double r189381 = r189378 + r189380;
        double r189382 = 1.0;
        double r189383 = r189381 - r189382;
        return r189383;
}
↓
double f(double a, double b) {
        double r189384 = a;
        double r189385 = 4.0;
        double r189386 = pow(r189384, r189385);
        double r189387 = b;
        double r189388 = pow(r189387, r189385);
        double r189389 = 2.0;
        double r189390 = pow(r189384, r189389);
        double r189391 = pow(r189387, r189389);
        double r189392 = r189390 * r189391;
        double r189393 = r189389 * r189392;
        double r189394 = r189388 + r189393;
        double r189395 = r189386 + r189394;
        double r189396 = 1.0;
        double r189397 = r189395 - r189396;
        return r189397;
}

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 2020018 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))