Average Error: 0.2 → 0.2
Time: 28.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) \cdot 4\right) - 1\]
\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(1 - 3 \cdot a\right)\right)\right) - 1
↓
\left({\left(a \cdot a + b \cdot b\right)}^{2} + \left(\left(a \cdot a\right) \cdot \left(a + 1\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) \cdot 4\right) - 1
double f(double a, double b) {
        double r4882867 = a;
        double r4882868 = r4882867 * r4882867;
        double r4882869 = b;
        double r4882870 = r4882869 * r4882869;
        double r4882871 = r4882868 + r4882870;
        double r4882872 = 2.0;
        double r4882873 = pow(r4882871, r4882872);
        double r4882874 = 4.0;
        double r4882875 = 1.0;
        double r4882876 = r4882875 + r4882867;
        double r4882877 = r4882868 * r4882876;
        double r4882878 = 3.0;
        double r4882879 = r4882878 * r4882867;
        double r4882880 = r4882875 - r4882879;
        double r4882881 = r4882870 * r4882880;
        double r4882882 = r4882877 + r4882881;
        double r4882883 = r4882874 * r4882882;
        double r4882884 = r4882873 + r4882883;
        double r4882885 = r4882884 - r4882875;
        return r4882885;
}
↓
double f(double a, double b) {
        double r4882886 = a;
        double r4882887 = r4882886 * r4882886;
        double r4882888 = b;
        double r4882889 = r4882888 * r4882888;
        double r4882890 = r4882887 + r4882889;
        double r4882891 = 2.0;
        double r4882892 = pow(r4882890, r4882891);
        double r4882893 = 1.0;
        double r4882894 = r4882886 + r4882893;
        double r4882895 = r4882887 * r4882894;
        double r4882896 = 3.0;
        double r4882897 = r4882896 * r4882886;
        double r4882898 = r4882893 - r4882897;
        double r4882899 = r4882889 * r4882898;
        double r4882900 = r4882895 + r4882899;
        double r4882901 = 4.0;
        double r4882902 = r4882900 * r4882901;
        double r4882903 = r4882892 + r4882902;
        double r4882904 = r4882903 - r4882893;
        return r4882904;
}

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

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

  3. Applied *-commutative0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))