Average Error: 0.2 → 0.0
Time: 28.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\left(4 \cdot \left(b \cdot b\right) + {\left(\sqrt{b \cdot b + a \cdot a}\right)}^{4}\right) - 1
double f(double a, double b) {
        double r2214856 = a;
        double r2214857 = r2214856 * r2214856;
        double r2214858 = b;
        double r2214859 = r2214858 * r2214858;
        double r2214860 = r2214857 + r2214859;
        double r2214861 = 2.0;
        double r2214862 = pow(r2214860, r2214861);
        double r2214863 = 4.0;
        double r2214864 = r2214863 * r2214859;
        double r2214865 = r2214862 + r2214864;
        double r2214866 = 1.0;
        double r2214867 = r2214865 - r2214866;
        return r2214867;
}
↓
double f(double a, double b) {
        double r2214868 = 4.0;
        double r2214869 = b;
        double r2214870 = r2214869 * r2214869;
        double r2214871 = r2214868 * r2214870;
        double r2214872 = a;
        double r2214873 = r2214872 * r2214872;
        double r2214874 = r2214870 + r2214873;
        double r2214875 = sqrt(r2214874);
        double r2214876 = pow(r2214875, r2214868);
        double r2214877 = r2214871 + r2214876;
        double r2214878 = 1.0;
        double r2214879 = r2214877 - r2214878;
        return r2214879;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied associate-*r*0.1

    \[\leadsto \left(\left(b \cdot b\right) \cdot 4 + \color{blue}{\left(\left(a \cdot a + b \cdot b\right) \cdot \sqrt{a \cdot a + b \cdot b}\right) \cdot \sqrt{a \cdot a + b \cdot b}}\right) - 1\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.1

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

  8. Applied pow30.1

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

  9. Applied pow-plus0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019149 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))