Average Error: 0.2 → 0.2
Time: 7.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\left(a \cdot a\right) \cdot \left(\sqrt[3]{1 - a} \cdot \sqrt[3]{1 - a}\right)\right) \cdot \sqrt[3]{1 - a} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(\left(a \cdot a\right) \cdot \left(\sqrt[3]{1 - a} \cdot \sqrt[3]{1 - a}\right)\right) \cdot \sqrt[3]{1 - a} + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1
double f(double a, double b) {
        double r221909 = a;
        double r221910 = r221909 * r221909;
        double r221911 = b;
        double r221912 = r221911 * r221911;
        double r221913 = r221910 + r221912;
        double r221914 = 2.0;
        double r221915 = pow(r221913, r221914);
        double r221916 = 4.0;
        double r221917 = 1.0;
        double r221918 = r221917 - r221909;
        double r221919 = r221910 * r221918;
        double r221920 = 3.0;
        double r221921 = r221920 + r221909;
        double r221922 = r221912 * r221921;
        double r221923 = r221919 + r221922;
        double r221924 = r221916 * r221923;
        double r221925 = r221915 + r221924;
        double r221926 = r221925 - r221917;
        return r221926;
}
↓
double f(double a, double b) {
        double r221927 = a;
        double r221928 = r221927 * r221927;
        double r221929 = b;
        double r221930 = r221929 * r221929;
        double r221931 = r221928 + r221930;
        double r221932 = 2.0;
        double r221933 = pow(r221931, r221932);
        double r221934 = 4.0;
        double r221935 = 1.0;
        double r221936 = r221935 - r221927;
        double r221937 = cbrt(r221936);
        double r221938 = r221937 * r221937;
        double r221939 = r221928 * r221938;
        double r221940 = r221939 * r221937;
        double r221941 = 3.0;
        double r221942 = r221941 + r221927;
        double r221943 = r221930 * r221942;
        double r221944 = r221940 + r221943;
        double r221945 = r221934 * r221944;
        double r221946 = r221933 + r221945;
        double r221947 = r221946 - r221935;
        return r221947;
}

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-cube-cbrt0.2

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

  4. Applied associate-*r*0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020036 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))