Average Error: 0.2 → 0.2
Time: 28.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
↓
\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} - 1
double f(double a, double b) {
        double r13169988 = a;
        double r13169989 = r13169988 * r13169988;
        double r13169990 = b;
        double r13169991 = r13169990 * r13169990;
        double r13169992 = r13169989 + r13169991;
        double r13169993 = 2.0;
        double r13169994 = pow(r13169992, r13169993);
        double r13169995 = 4.0;
        double r13169996 = 1.0;
        double r13169997 = r13169996 - r13169988;
        double r13169998 = r13169989 * r13169997;
        double r13169999 = 3.0;
        double r13170000 = r13169999 + r13169988;
        double r13170001 = r13169991 * r13170000;
        double r13170002 = r13169998 + r13170001;
        double r13170003 = r13169995 * r13170002;
        double r13170004 = r13169994 + r13170003;
        double r13170005 = r13170004 - r13169996;
        return r13170005;
}
↓
double f(double a, double b) {
        double r13170006 = a;
        double r13170007 = r13170006 * r13170006;
        double r13170008 = b;
        double r13170009 = r13170008 * r13170008;
        double r13170010 = r13170007 + r13170009;
        double r13170011 = 2.0;
        double r13170012 = pow(r13170010, r13170011);
        double r13170013 = 3.0;
        double r13170014 = r13170006 + r13170013;
        double r13170015 = r13170014 * r13170009;
        double r13170016 = 1.0;
        double r13170017 = r13170016 - r13170006;
        double r13170018 = r13170007 * r13170017;
        double r13170019 = r13170015 + r13170018;
        double r13170020 = 4.0;
        double r13170021 = r13170019 * r13170020;
        double r13170022 = r13170012 + r13170021;
        double r13170023 = sqrt(r13170022);
        double r13170024 = r13170023 * r13170023;
        double r13170025 = r13170024 - r13170016;
        return r13170025;
}

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

    \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.2

    \[\leadsto \color{blue}{\sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)}} - 1\]

  4. Final simplification0.2

    \[\leadsto \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} \cdot \sqrt{{\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a + 3\right) \cdot \left(b \cdot b\right) + \left(a \cdot a\right) \cdot \left(1 - a\right)\right) \cdot 4} - 1\]

Reproduce

herbie shell --seed 2019169 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (- 1.0 a)) (* (* b b) (+ 3.0 a))))) 1.0))